An Introduction to Matched Asymptotic Expansions(A Draft)

Peicheng Zhu

Basque Center for Applied Mathematics andIkerbasque Foundation for Science

Nov., 2009

Preface

The Basque Center for Applied Mathematics (BCAM) is a newly foundedinstitute which is one of the Basque Excellence Research Centers (BERC).BCAM hosts a series of training courses on advanced aspects of applied andcomputational mathematics that are delivered, in English, mainly to graduatestudents and postdoctoral researchers, who are in or from out of BCAM. Theseries starts in October 2009 and finishes in June 2010. Each of the 13 coursesis taught in one week, and has duration of 10 hours.

This is the lecture notes of the course on Asymptotic Analysis that I gaveat BCAM from Nov. 9 to Nov. 13, 2009, as the second of this series of BCAMcourses. Here I have added some preliminary materials. In this course, weintroduce some basic ideas in the theory of asymptotic analysis. Asymptoticanalysis is an important branch of applied mathematics and has a broad rangeof contents. Thus due to the time limitation, I concentrate mainly on themethod of matched asymptotic expansions. Firstly some simple examples,ranging from algebraic equations to partial differential equations, are discussedto give the reader a picture of the method of asymptotic expansions. Then weend with an application of this method to an optimal control problem, whichis concerned with vanishing viscosity method and alternating descent methodfor optimal control of scalar conservation laws in presence of non-interactingshocks.

Peicheng ZhuBilbao, Spain

i

Jan., 2010

Contents

Preface i

1 Introduction 1

2 Algebraic equations 52.1 Regular perturbation . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Iterative method . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Singular perturbation . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Non-ingeral powers . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Ordinary differential equations 133.1 First order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Regular . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Singular . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.2.1 Outer expansions . . . . . . . . . . . . . . . . . 163.1.2.2 Inner expansions . . . . . . . . . . . . . . . . . 173.1.2.3 Matched asymptotic expansions . . . . . . . . . 18

3.2 Second order ODEs and boundary layers . . . . . . . . . . . . . 193.2.1 Outer expansions . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Inner expansions . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2.1 Rescaling . . . . . . . . . . . . . . . . . . . . . 223.2.3 Matching conditions . . . . . . . . . . . . . . . . . . . . 24

3.2.3.1 Matching by expansions . . . . . . . . . . . . . 253.2.3.2 Van Dyke’s rule for matching . . . . . . . . . . 26

3.2.4 Matched asymptotic expansions . . . . . . . . . . . . . . 273.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Partial differential equations 294.1 Regular problem . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Conservation laws and vanishing viscosity method . . . . . . . . 29

4.2.1 Construction of approximate solutions . . . . . . . . . . 30

iii

4.2.1.1 Outer and inner expansions . . . . . . . . . . . 304.2.1.2 Matching conditions and approximations . . . . 30

4.2.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 An application to optimal control theory 315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Sensitivity analysis: the inviscid case . . . . . . . . . . . . . . . 36

5.2.1 Linearization of the inviscid equation . . . . . . . . . . . 365.2.2 Sensitivity in presence of shocks . . . . . . . . . . . . . . 395.2.3 The method of alternating descent directions: Inviscid

case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Matched asymptotic expansions and approximate solutions . . . 46

5.3.1 Outer expansions . . . . . . . . . . . . . . . . . . . . . . 475.3.2 Derivation of the interface equations . . . . . . . . . . . 525.3.3 Inner expansions . . . . . . . . . . . . . . . . . . . . . . 555.3.4 Approximate solutions . . . . . . . . . . . . . . . . . . . 58

5.4 Convergence of the approximate solutions . . . . . . . . . . . . 605.4.1 The equations are satisfied asymptotically . . . . . . . . 615.4.2 Proof of the convergence . . . . . . . . . . . . . . . . . . 67

5.5 The method of alternating descent directions: Viscous case . . . 74Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Bibliography 81

Index 84

Chapter 1

Introduction

In the real world, many problems (which arise in applied mathematics, physics,engineering sciences, · · · , also pure mathematics like the theory of numbers)don’t have a solution which can be written a simple, exact, explicit formula.Some of them have a complex formula, but we don’t know too much aboutsuch a formula.

We now consider some examples. i) The Stirling formula:

n! ∼√2nπe−nnn

(1 +O

(1

n

)). (1.0.1)

The Landau symbol the big Oh “O” and the Du Bois Reymond symbol “∼”are used. Note that n! grows very quickly as n → ∞ and becomes so largethat one can not have any idea about how big it is. But formula (1.0.1) givesus a good estimate of n!.

ii) From Algebra we know that in general there is no explicit solution to analgebraic equation with degree n ≥ 5.

iii) Most of problems in the theory of nonlinear ordinary or partial differen-tial equations don’t have an exact solution.

And many others.

In practice, however, an approximation of a solution to such problems isusually enough. Thus the approaches to finding such an approximation isimportant. There are two main methods. One is numerical approximation,which is especially powerful after the invention of the computer and is nowregarded as the third most import method for scientific research (just after thetraditional two: theoretical and experimental methods). Another is analyti-cal approximation with an error which is understandable and controllable, inparticular, the error could be made smaller by some rational procedure. Theterm “analytical approximate solution” means that an analytic formula of anapproximate solution is found and its difference with the exact solution.

1

Asymptotic analysis is powerful tool for finding analytical approximate so-lutions to complicated practical problems, which is an important branch ofapplied mathematics. In 1886 the establishment of rigorous foundation wasdone by Poincare and Stieltjes. They published separately papers on asymp-totic series. Later in 1905, Prandtl published a paper on the motion of a fluidor gas with small viscosity along a body. In the case of an airfoil movingthrough air, such problem is described by the Navier-Stokes equations withlarge Reynolds number. The method of singular perturbation was proposed.

Of course, the history of asymptotic analysis can be traced back to muchearlier than 1886, even to the time when our ancestors studied the problem,as small as the measure of a rod, or as large as the study of the perturbedorbit of a planet. As we know, when we measure a rod, each measure gives adifferent value, so n-measures result in n-different values. Which one shouldchoose to be the length of this rod? The best approximation to the real lengthof the rod is the mean value of these n-numbers, and each of the measures canbe regarded as a perturbation of the mean value.

The Sun’s gravitational attraction is the main force acting on each planet,but there are much weaker gravitational forces between the planets, whichproduce perturbations of their elliptical orbits; these make small changes ina planet’s orbital elements with time. The planets which perturb the Earth’sorbit most are Venus, Jupiter, and Saturn. These planets and the sun alsoperturb the Moon’s orbit around the Earth-Moon system’s center of mass.The use of mathematical series for the orbital elements as functions of timecan accurately describe perturbations of the orbits of solar system bodies forlimited time intervals. For longer intervals, the series must be recalculated.

Today, astronomers use high-speed computers to figure orbits in multiplebody systems such as the solar system. The computers can be programmedto make allowances for the important perturbations on all the orbits of themember bodies. Such calculations have now been made for the Sun and themajor planets over time intervals of up to several tens of millions of years.

As accurately as these calculations can be made, however, the behavior ofcelestial bodies over long periods of time cannot always be determined. Forexample, the perturbation method has so far been unable to determine thestability either of the orbits of individual bodies or of the solar system as awhole for the estimated age of the solar system. Studies of the evolution ofthe Earth-Moon system indicate that the Moon’s orbit may become unstable,which will make it possible for the Moon to escape into an independent orbitaround the Sun. Recent astronomers have also used the theory of chaos toexplain irregular orbits.

The orbits of artificial satellites of the Earth or other bodies with atmos-pheres whose orbits come close to their surfaces are very complicated. The

orbits of these satellites are influenced by atmospheric drag, which tends tobring the satellite down into the lower atmosphere, where it is either vaporizedby atmospheric friction or falls to the planet’s surface. In addition, the shapeof Earth and many other bodies is not perfectly spherical. The bulge thatforms at the equator, due to the planet’s spinning motion, causes a strongergravitational attraction. When the satellite passes by the equator, it may beslowed enough to pull it to earth.

The above argument gives us many problems with small perturbations, someof those perturbations can be omitted under suitable assumptions.

The main contents of asymptotic analysis are as follows: perturbation me-thod, the method of multi-scale expansions, averaging method, WKBJ (Went-zel, Kramers, Brillouin and Jeffreys) approximation, the method of matchedasymptotic expansions, asymptotic expansion of integrals, and so on. Thiscourse is mainly concerned with the method of matched asymptotic expan-sions. Firstly we study some simple examples arising in algebraic equation,ordinary differential equations, from which we will get key ideas of matchedasymptotic expansion, though those examples are simple. Then we shall in-vestigate matched asymptotic expansion for partial differential equations andfinally take an optimal control problem as an application.

Let us now introduce some notations. D ⊂ Rd with d ∈ N denotes an opensubset in Rd. f, g, h : D → R are real continuous functions. We denote a smallquantity by ε. The Landau symbols the big Oh O and the little o will be used.

Definitions. A sequence of gauge functions φn(x) (n = 0, 1, 2, · · · ) is saidto be form an asymptotic sequence as x→ x0 if, for all n,

φn+1(x) = o(φn(x)), as x→ x0.

If φn(x) is an asymptotic sequence of gauge functions as x → x0, we saythat

∞∑n=1

anφn(x), an are constant function,

is an asymptotic expansion (or asymptotic approximation) of a function f(x)as x→ x0, if for each N .

f(x) =N∑

n=1

anφn(x) + o(φN(x)), as x→ x0.

Chapter 2

Algebraic equations

In this chapter we shall investigate some algebraic equations, which are veryhelpful for establishing the picture of asymptotic analysis in our mind, thoughthose examples are quite simple. Let us consider algebraic equations with asmall positive parameter which is denoted by ε in what follows.

2.1 Regular perturbation

Consider the following quadratic equation

x2 − εx− 1 = 0. (2.1.1)

Suppose that ε = 0, equation (2.1.1) becomes

x2 − 1 = 0. (2.1.2)

It is easy to find the roots xε of (2.1.1), for any fixed ε, which read

xε1 =ε+

√ε2 + 4

2, and xε2 =

ε−√ε2 + 4

2. (2.1.3)

Correspondingly, the roots x0 of (2.1.2) are x01,2 = ±1.A natural question arises: Does xε converge to x0? We prove easily that

xε1 → x1, and xε2 → x2. (2.1.4)

So (2.1.1) is called a regular perturbation of (2.1.2). The perturbation term is−εx. A perturbation is called singular if it is not regular.

For most of practical problems, however, we don’t have the explicit formulaslike (2.1.1). In this case, how can we get the knowledge about the limits asthe small parameter ε goes to zero?

5

The method of asymptotic expansions is a powerful tool for such an inves-tigation. Note that xε depends on ε, to construct an asymptotic expansion,we define an ansatz as follows

xε = εα0(x0 + εα1x1 + εα2x2 + · · · ), (2.1.5)

here, αi (i = 0, 1, 2, · · · ) are constant to be determined, and we assume, withoutloss of generality, that x0, x1, differ from zero, and 0 < α1 < α2 < · · · .

We first determine α0. There are three cases. i) α0 > 0, ii) α0 < 0 and iii)α0 = 0. We will show that only case iii) is possible to get an asymptotic ex-pansion. Inserting ansatz (2.1.5) into equation (2.1.1). Balancing the resultingequation, we obtain

x2α0x20 + 2ε2α0+α1x0x1 + ε2(α0+α1)x21 − xα0x0 − εα0+α1x1 − 1 + · · · ) = 0.(2.1.6)

Suppose now that case i) happens, i.e. α0 > 0, which implies α0 is thesmallest power. Thus from (2.1.6) it follows that the coefficient of εα0 shouldbe zero, namely x0 = 0, this is contradict to our assumption that x0 = 0.

For case ii), namely α0 < 0, we have 2α0 < α0 < α0 + α1, thus 2α0 is thesmallest power and the coefficient of ε2α0 should be zero, so x20 = 0, whichviolates our assumption too.

Therefore, we assert that only the case α0 = 0 is possible, and (2.1.5)becomes

xε = x0 + εα1x1 + εα2x2 + · · · , (2.1.7)

moreover, (2.1.6) now is

x20 + 2εα1x0x1 + ε2α1x21 − x0 − εα1x1 − 1 + · · · ) = 0. (2.1.8)

Similar to the above procedure for deciding α0, we can determine α1, α2

etc., which are α1 = 1, α2 = 2, · · · . So ansatz (2.1.5) takes the following form

xε = x0 + ε1x1 + ε2x2 + · · · , (2.1.9)

and the following expansions are obtained

ε0 : x20 − 1 = 0, (2.1.10)

ε1 : 2x0x1 − x0 = 0, (2.1.11)

ε2 : 2x0x2 + x21 − x1 = 0. (2.1.12)

Solving (2.1.10) we have x0 = 1 or x0 = −1. We take the first case as anexample and construct the asymptotic expansion. From (2.1.11) and (2.1.12)we get, respectively,

x1 =1

2, x2 =

1

8.

Up to i-terms (i = 1, 2, 3), we expand xε as follows

Xε1 = 1, (2.1.13)

Xε2 = 1 +

ε

2, (2.1.14)

Xε3 = 1 +

ε

2+ε2

8. (2.1.15)

The next question then arise: How precisely does Xεi (i = 1, 2, 3) satisfy the

equation of xε?Straightforward computations yield that

(Xε1)

2 − εXε1 − 1 = O(ε), (2.1.16)

(Xε2)

2 − εXε2 − 1 = O(ε2), (2.1.17)

(Xε3)

2 − εXε3 − 1 = O(ε4). (2.1.18)

From which it is easy to see that Xεi satisfies very well the equation when ε is

small, and the error becomes smaller as i is larger, which means that we takemore terms.

2.2 Iterative method

In this section we are going to make use the so called iterative method toconstruct asymptotic expansions again for equation (2.1.1). We rewrite (2.1.1)as follows

x = ±√1 + εx,

where x = xε. This formula suggests us an iterative procedure,

xn+1 =√1 + εxn (2.2.1)

for any n ∈ N. Here we only take the positive root as an example. Let x0 bea fixed real number. One then obtains from (2.2.1) that

x1 = 1 +ε

2x0 + · · · , (2.2.2)

so we find the first term of an asymptotic expansion, however the secondterm in (2.2.2) still depends on x0. To get the second term of an asymptoticexpansion, we iterate once again and arrive at

x2 = 1 +ε

2x1 + · · · = 1 +

ε

2+ · · · , (2.2.3)

this gives us the desired result. After iterating twice, we then construct anasymptotic expansion:

xε = 1 +ε

2+ · · · . (2.2.4)

The shortage of this method for constructing an asymptotic expansion isthat we don’t have an explicit formula, like (2.2.1), which guarantees the ite-ration converges.

2.3 Singular perturbation

We now investigate the following equation which will give us very differentresults.

εx2 − x− 1 = 0. (2.3.1)

Suppose that ε = 0, equation (2.3.1) becomes

−x− 1 = 0. (2.3.2)

Therefore we see that one root of (2.3.1) disappears as ε becomes 0. This isvery different from (2.1.1). It is not difficult to get the roots of (2.3.1) whichread

xε =1±

√1 + 4ε

2ε.

There hold, as ε→ 0, that

xε− =1−

√1 + 4ε

2ε→ −1,

and by a Taylor expansion,

xε+ =1 +

√1 + 4ε

2ε

=1

2ε

(1 + 1 + 2ε− 2ε2 + · · ·

)=

1

ε+ 1− 2ε+ · · · . (2.3.3)

Therefore we see that one root i.e. xε− converges to that of the limit equation(2.3.2), another blows up at the rate 1

ε. Thus we can not expect that an

asymptotic expansion like we did for a regular problem is valid too in thiscase. How to find a suitable scale for such a problem? We shall make use ofthe rescaling technique.

2.3.1 Rescaling

Suppose that we don’t know a priori the correct scale for constructing anasymptotic expansion, then the rescaling technique helps us to find it. Thissubsection is concerned with it, and we take (2.3.1) as an example.

Let δ be a real function in ε, and let

x = δX,

where δ = δ(ε) and X = O(1). A rescaling technique is to determine thefunction δ, consequently, a new variable X is found.

Rewriting (2.3.1) in X, we have

εδ2X2 − δX − 1 = 0. (2.3.4)

By comparing the coefficients of (2.3.4), namely,

εδ2, δ, 1,

we divide the rescaling argument into five cases.

Case i) δ << 1. Then (2.3.4) can be written as

1 = εδ2X2︸ ︷︷ ︸o(1)

− δX︸︷︷︸o(1)

= o(1), (2.3.5)

which can not be true since the left hand side of (2.3.5) while the right handside is a very small quantity.

Case ii) δ = 1, which means there is no any change to (2.3.1). (2.3.4) becomes

εX2︸︷︷︸o(1)

−X − 1 = 0, (2.3.6)

thus it is impossible that X = 1 and we can construct a regular asymptoticexpansion but cannot recover the lost root.

Case iii) 1 << δ << 1εwhich implies that δε << 1. Dividing equation (2.3.4)

by δ we obtain

εδX2︸ ︷︷ ︸o(1)

−X − 1

δ︸︷︷︸o(1)

= 0, (2.3.7)

and X = o(1). This is impossible.

Case iv) δ = 1ε, namely δε = 1, also δ >> 1 since we assume that ε << 1.

Consequently, we infer from (2.3.4) that

X2 −X − 1

δ︸︷︷︸o(1)

= 0, (2.3.8)

Thus X ∼ 0 or 1. This gives us the correct scale.

Case v) δ >> 1ε, thus δε >> 1. Multiplying (2.3.4) by ε−1δ−2 yields

X2 − (εδ)−1X︸ ︷︷ ︸o(1)

− 1

εδ2︸︷︷︸o(1)

= 0, (2.3.9)

and X = o(1). So this is not a suitable scale either.

In conclusion, the suitable scale is δ = 1ε, thus

x =X

ε.

(2.3.4) is turn out to be

X2 −X − ε = 0. (2.3.10)

A singular problem is reduced to a regular one.

We now turn back to equation (2.3.1). Rescaling suggests us to use thefollowing ansatz:

xε = ε−1x−1 + x0 + εx1 + · · · . (2.3.11)

Inserting into (2.3.1) and comparing the coefficients of ε1 (i = −1, 0, 1, · · · )on both sides of (2.3.1), we obtain

ε−1 : x2−1 − x−1 = 0, (2.3.12)

ε0 : 2x0x−1 − x0 − 1 = 0, (2.3.13)

ε1 : x20 + 2x−1x1 − x1 = 0. (2.3.14)

The roots of (2.3.12) are x−1 = 1 and x−1 = 0. The second root does not yielda singular asymptotic expansion, thus it can be excluded easily. We considernow x−1 = 1. From (2.3.13) and (2.3.14) one solves

x0 = 1, x1 = −1.

Therefore, we construct approximations Xεi i = 0, 1, 2, of the root xε+ by

Xε2 =

1

ε+ 1− ε, (2.3.15)

Xε1 =

1

ε+ 1, (2.3.16)

and

Xε0 =

1

ε. (2.3.17)

How precisely do they satisfy equation (2.3.1)? Computations yield

ε(Xε0)

2 −Xε0 − 1 = −1,

correspondingly, xε+ −Xε0 = 1− 2ε+ o(ε) → 0. This means that an expansion

with one term is not a good approximation.

ε(Xε1)

2 −Xε1 − 1 = ε,

and

ε(Xε2)

2 −Xε2 − 1 = O(ε2),

meanwhile, xε+ −Xε1 = O(ε), and xε+ −Xε

2 = O(ε2).Thus Xε

1 , Xε2 are a good approximation to xε+. Moreover, we can conclude

that the more terms we take, the more precise the approximation is. We alsofigure out the profile approximately of xε+, in other word, we know now howthe root disappears by blowing up as ε goes to zero.

2.4 Non-ingeral powers

Any of the asymptotic expansions in Sections 1.1 and 1.2 are a series withintegral powers. However this is in general not true. Here we give an example.Consider

(1− ε)x2 − 2x+ 1 = 0. (2.4.1)

Define, as in previous sections, an ansatz as follows

xε = x0 + εx1 + εx2 + · · · , (2.4.2)

inserting (2.4.2) into (2.4.1) and balancing both sides yield

ε0 : x20 − 2x0 + 1 = 0, (2.4.3)

ε0 : 2x0x1 − 2x1 − x20 = 0, (2.4.4)

ε1 : 2x0x2 − 2x2 + x21 − 2x0x1 = 0. (2.4.5)

From (2.4.3) one gets x0 = 1, whence (2.4.4) implies 2x1 − 2x1 − 12 = 0,that is 1 = 0, a contradiction. So (2.4.2) is not well-defined.

Now we define an ansatz as

xε = x0 + εαx1 + εβx2 + · · · , (2.4.6)

where 0 < α < β < · · · are constants to be determined. inserting this ansatzinto (2.4.1) and balancing both sides, we obtain there must hold that

α =1

2, β = 1, · · · .

and the correct ansatz is

xε = x0 + ε12x1 + εx2 + ε

32x3 + · · · . (2.4.7)

The remaining part of construction is similar to previous ones. We have xε =1± ε

12 + ε± ε

32 + · · · .

Chapter 3

Ordinary differential equations

We start this chapter with some definitions. Consider

L0[u] + εL1[u] = f0 + εf1, in D. (3.0.1)

and the associated equation corresponding to the case that ε = 0

L0[u] = f0. in D. (3.0.2)

Here, L0, L1 are known operators, either ordinary or partial; f0, f1 are givenfunctions.

The terms εL1[u] and εf1 are called perturbations.Eε (E0 respectively) denotes the problem consisting equation (3.0.1) ((3.0.2)

respectively) and suitable boundary/initial conditions. The solution to pro-blem Eε (or E0) is denoted by uε (or u0).

Definition. Problem Eε is regular if

∥uε − u0∥D → 0,

as ε→ 0. Otherwise, Problem Eε is referred to a singular one.Here ∥·∥D is a suitable norm over domain D. Note that a problem is regular

or singular depends on the choice of the norm, which can be clarified by thefollowing problem.

Example. Let D = (0, 1). A real function φ : D → R is a solution to

εd2φ

dx2+dφ

dx= 0, in D, (3.0.3)

φ|x=0 = 0, φ|x=1 = 1. (3.0.4)

The solution φ is

φ = φ(x; ε) =1− e−

xε

1− e−1ε

,

13

which is monotone increasing.Now we define two norms. i) ∥φ∥D = maxD |φ|, then problem (3.0.3) –

(3.0.4) is singular since ∥φ− φ0∥D = 1 where φ0 = 0 or φ0 = 1.ii) Define

∥φ∥D =

(∫D

|φ|2) 1

2

,

choose φ0 = 1 which satisfies dφdx

= 0, then we can prove easily that ∥φ −φ0∥D → 0 as ε→ 0, whence problem (3.0.3) – (3.0.4) is regular.

In what follows, we restrict ourself to consider only the maximum norm, andin the remaining part of this chapter the domain D is defined by D = (0, 1).

3.1 First order ODEs

3.1.1 Regular

In this subsection we first consider a regular problem of ordinary differentialequations of first order. Consider

du

dx+ u = εx, in D. (3.1.1)

u(0) = 1, (3.1.2)

and its associated problem

du

dx+ u = 0, in D. (3.1.3)

u(0) = 1, (3.1.4)

We can solve easily these problems whose solution reads

uε(x) = (1 + ε)e−x + ε(x− 1), u0(x) = e−x. (3.1.5)

Calculating the difference of these two solutions yields

∥uε − u0∥D = εmaxD

|e−x + x− 1| → 0

as ε→ 0. Therefore, problem (3.1.1) – (3.1.2) is regular, and the term εx is aregular perturbation.

But, in general, one can not expect that there are the explicit simple for-mulas, like (3.1.5), of exact solutions. Thus we next deal with this problem in

a general way, and employ the method of asymptotic expansions. To this end,we define an ansatz

uε(x) = u0(x) + εu1(x) + ε2u2(x) + · · · , (3.1.6)

insert it into equation (3.1.1) to get

ε0 : u′0 + u0 = 0, u0(0) = 1, (3.1.7)

ε0 : u′1 + u1 = x, u1(0) = 0, (3.1.8)

ε1 : u′2 + u2 = 0, u2(0) = 0. (3.1.9)

The condition u0(0) = 1 follows from (3.1.2) and ansatz (3.1.6). In fact, thereholds

1 = uε(0) = u0(0) + εu1(0) + ε2u2(0) + · · · → u0(0), (3.1.10)

thus, u0(0) = 1. With this in hand, we use again (3.1.10) to derive the condi-tion u1(0) = 0. We obtain

0 = εu1(0) + ε2u2(0) + · · · , whence0 = u1(0) + εu2(0) + · · · (3.1.11)

Letting ε → 0 we get u1(0) = 0. In a similar manner, we derive the conditionu2(0) = 0.

Solving problems (3.1.7), (3.1.8) and (3.1.9) we have

u0(x) = e−x, u1(x) = x− 1 + e−x, u2(x) = 0. (3.1.12)

Thus the approximations can be constructed:

U ε0 (x) = e−x, (3.1.13)

U ε1 (x) = U ε

2 (x) = (1 + ε)e−x + ε(x− 1). (3.1.14)

Simple calculation shows that U ε0 (x) satisfies (3.1.1) with a small error

(U ε1 (x))

′ + U ε1 (x)− εx = O(ε),

and condition (3.1.2) is satisfied exactly. Note that U ε1 (x) = U ε

2 (x) are equalto the exact solution (3.1.5), so they solve problem (3.1.1) – (3.1.2), and are avery good “approximation”.

3.1.2 Singular

Now we are going to study singular perturbation and boundary layers. Theperturbed problem is

εdu

dx+ u = x, in D. (3.1.15)

u(0) = 1, (3.1.16)

and the associated one is

u = x, in D. (3.1.17)

u(0) = 1, (3.1.18)

The exact solutions of problem (3.1.15) – (3.1.16) is

uε(x) = (1 + ε)e−x + x− ε, (3.1.19)

Let u0(x) = x. Computation yields

∥uε − u0∥D = maxD

|(1 + ε)e−x − ε| = 1,

for any positive ε. Therefore, by definition, problem (3.1.15) – (3.1.16) issingular, and the term εdu

dxis a singular perturbation.

We next want to employ the method of asymptotic expansions to studythis singular problem, for such a problem at least one boundary layer arisesand a matched asymptotic expansion is suitable for it. We will constructouter and inner expansions which are valid in the so-called outer and innerregions, respectively. Then we derive matching conditions which are enableus to establish an asymptotic expansion which is valid uniformly in the wholedomain. Thus we start with the construction of outer expansions.

3.1.2.1 Outer expansions

The ansatz for deriving outer expansion is just of the form of a regular expan-sion:

uε(x) = u0(x) + εu1(x) + ε2u2(x) + · · · , (3.1.20)

Similar to the approach for asymptotic expansion of the regular problem(3.1.1) – (3.1.2), we obtain

ε0 : u0(x) = x, (3.1.21)

ε1 : u1(x) + u′0(x) = 0, (3.1.22)

ε2 : u2(x) + u′1(x) = 0. (3.1.23)

Solving the above problems yields

u0(x) = x, (3.1.24)

u1(x) = −1, (3.1.25)

u2(x) = 0. (3.1.26)

Then we get approximations:

Oε2(x) = x− ε. (3.1.27)

Moreover, from (3.1.24) we obtain that u0(0) = 0 which differs from thegiven condition u0(0) = 1, thus there is a boundary layer appearing at x = 0.

3.1.2.2 Inner expansions

To construct inner expansions, we introduce a new variable, the so-called fastvariable:

ξ =x

ε,

On one hand, for a beginner to the theory of asymptotic analysis, it may benot easy to understand why we define ξ in this form, the power of ε is 1? Toconvince oneself, one may assume a more general form as ξ = x

εαwith α ∈ R.

Then repeating the procedure we will carry out later in next subsection, weprove that α must be equal to 1 in order to get an asymptotic expansion. Onthe other hand we already assume that a boundary layer (Definition!) occurs atx = 0. If the assumption is incorrect, the procedure will break down when youtry to match the inner and outer expansions in the intermediate region. At thispoint one may assume that there exists a boundary layer near a point x = x0.The following analysis is the same, except that the scale transformation in theboundary layer is ξ = x−x0

εδ. We shall carry out the analysis for determining δ

in the next section.An inner expansion is in terms of ξ, we assume that

uε(x) = U0(ξ) + εU1(ξ) + ε2U2(ξ) + · · · . (3.1.28)

It is easy to compute that for i = 0, 1, 2, · · · ,

dUi(ξ)

dx=

1

ε

dUi(ξ)

dξ.

Invoking equation (3.1.1) we arrive at

ε−1 : U ′0 + U0 = 0, (3.1.29)

ε0 : U ′1 + U1 = ξ, (3.1.30)

ε1 : U ′2 + U2 = 0. (3.1.31)

From which we have

U0(ξ) = C0e−ξ, U1(ξ) = C1e

−ξ + ξ − 1, U2(ξ) = C2e−ξ. (3.1.32)

Next step is to determine the constants Ci with i = 0, 1, 2. To this end, weuse the condition at x = 0 which implies that ξ = 0 too to conclude thatU0(0) = 1, thus C0 = 1. Similarly we have C1 = 1, C2 = 0. Therefore, aninner expansion can be obtained

Iε2(ξ) = (1 + ε)e−ξ + ε(ξ − 1). (3.1.33)

3.1.2.3 Matched asymptotic expansions

There are two main approaches to combine together the inner and outer ex-pansions. The first one is to take the sum of the inner expansion (3.1.33) andthe outer expansion (3.1.27), then subtract their common part which is validin the intermediate region. To get a matched asymptotic expansion, it remainsto find the common part. We start with

U0(ξ) + εU1(ξ) + ε2U2(ξ) = u0(x) + εu1(x) + ε2u2(x) +O(ε3).

Following Fife [14], we rewrite x = εξ and expand the right hand side interms of ξ. There hold

U0(ξ) + εU1(ξ) + ε2U2(ξ) = u0(εξ) + εu1(εξ) + ε2u2(εξ) +O(ε3)

= u0(0) + u′0(0)εξ +1

2u′′0(0)(εξ)

2

+ε(u1(0) + u′1(0)εξ) + u2(0) +O(ε3)

= u0(0) + ε(u′0(0)ξ + u1(0))

+ε2(1

2u′′0(0)ξ

2 + u′1(0)ξ) + u2(0)

)+O(ε3).(3.1.34)

Therefore we obtain the following matching conditions

U0(ξ) = u0(0) = 0, (3.1.35)

U1(ξ) ∼ u′0(0)ξ + u1(0) = ξ − 1, (3.1.36)

U2(ξ) ∼ 1

2u′′0(0)ξ

2 + u′1(0)ξ + u2(0) = 0, (3.1.37)

for ξ → ∞.The common part is U0(ξ) + εU1(ξ) + ε2U2(ξ) = ε(ξ − 1). The matched

asymptotic expansion then is

U ε2 (x) = Iε2(ξ) +Oε

2(x)− Common part

= (1 + ε)e−ξ + ε(ξ − 1) + (x− ε)− ε(ξ − 1)

= (1 + ε)e−ξ + x− ε. (3.1.38)

So U ε2 (x) is just the exact solution to problem (3.1.15) – (3.1.16).

The second method for constructing a matched asymptotic expansion frominner and outer expansions is to make use of a suitable cut-off function to forma linear combination of inner and outer expansions.

We define a function χ = χ(ξ) : R → R+ which is smooth such that

χ(ξ) =

1 if |ξ| ≤ 1,

0 if |ξ| ≥ 2,(3.1.39)

and 0 ≤ χ(ξ) ≤ 1 if ξ ∈ [1, 2]. And let

χε(x) = χ(ε−γx), (3.1.40)

which is easily seen that

supp(χε) ⊂ [0, 2εγ], supp(χ′ε), supp(χ

′′ε) ⊂ [εγ, 2εγ].

Here γ ∈ (0, 1) is a fixed number.Now we are able to define an approximation by

U ε2 (x) = (1− χε(x))O

ε2(ξ) + χε(x)I

ε2(x). (3.1.41)

By this method, we don’t need to find the common part and the argumentis simpler, however, the price we should pay is that U ε

2 (x) does not satisfyequation (3.1.15) precisely any more. Instead, an error occurs:

εdU ε

2 (x)

dx+ U ε

2 (x)− x = χ′ε(x))(I

ε2(ξ)−Oε

2(x))ε1−γ

= O(ε1−γ). (3.1.42)

3.2 Second order ODEs and boundary layers

The previous examples have given us some ideas about the method of asymp-totic expansions. However, they can not outline all the features of this methodsince they are really too simple. In this section, we are going to study amore complex problem which possesses all aspects of asymptotic expansions.Consider the problem for second order ordinary differential equation:

εd2u

dx2+ (1 + ε)

du

dx+ u = 0, in D. (3.2.1)

u(0) = 0, u(1) = 1. (3.2.2)

First of all, we explain that problem (3.2.1) – (3.2.2) is not regular. We set

uε(x) = u0(x) + εu1(x) + ε2u2(x) + · · · . (3.2.3)

Inserting into equation (3.2.1) and comparing the coefficients of εi in bothsides, one has

du0dx

+ u0 = 0, (3.2.4)

u0(0) = 0, u0(1) = 1. (3.2.5)

Then we further get

du1dx

+ u1 +d2u0dx2

+du0dx

= 0, (3.2.6)

u1(0) = 0. (3.2.7)

du2dx

+ u2 +d2u1dx2

+du1dx

= 0, (3.2.8)

u2(0) = 0. (3.2.9)

Since equation (3.2.4) is of first order, only one of conditions (3.2.5) can besatisfied. The solution to (3.2.4) is

u0(x) = C0e−x.

We shall see that even we require u0(x) meets only one condition in (3.2.5),there still is no asymptotic expansion of the form (3.2.3). There are two cases.

Case i) Suppose that the condition at x = 0 is satisfied (we don’t care at thismoment another condition), then C0 = 0, hence u0(x) = 0. Solving problems(3.2.6) – (3.2.7) and (3.2.8) – (3.2.9) one has

u1(x) = u2(x) = 0.

Thus no asymptotic expansion can be found.

Case ii) Assume that the condition at x = 1 is satisfied, then C0 = e, andu0(x) = e1−x. Consequently, equations (3.2.6) and (3.2.8) become

du1dx

+ u1 = 0,du2dx

+ u2 = 0. (3.2.10)

Hence,u1(x) = C1e

−x, u2(x) = C2e−x.

But from condition u1(1) = u2(1) = 0 which can be derived from ansatz (3.2.3),it follows that C1 = C2 = 0, whence

u1(x) = u2(x) = 0,

and the “possible” asymptotic expansion is U εi (x) = e1−x for any i = 0, 1, 2, · · · .

Note that |U εi (0)− uε(0)| = e → 0.

Therefore, (3.2.1) – (3.2.2) is singular.

3.2.1 Outer expansions

This subsection is concerned with outer expansions. We begin with the defi-nition of an ansatz

uε(x) = u0(ξ) + εu1(ξ) + ε2u2(ξ) + · · · . (3.2.11)

For simplicity of notations, we will denote the derivative of a one-variablefunction by ′, namely, f ′(x) = df

dx, f ′(ξ) = df

dξ, etc. Inserting (3.2.11) into

equation (3.2.1) and equating the coefficients of εi of both sides yield

ε0 : u′0 + u0 = 0, u0(1) = 1, (3.2.12)

ε1 : u′1 + u1 + u′′0 + u′0 = 0, u1(1) = 0, (3.2.13)

ε2 : u′2 + u2 + u′′1 + u′1 = 0, u2(1) = 0. (3.2.14)

The solutions to (3.2.12), (3.2.13) and (3.2.14) are respectively,

u0(x) = e1−x, u1(x) = u2(x) = 0.

Thus outer approximations (up to i+1-terms) can be constructed as follows

Oεi (x) = e1−x, (3.2.15)

here i = 0, 1, 2.

3.2.2 Inner expansions

The construction of an inner expansion is more complicated than that foran outer expansion. Firstly a correct scale should be decided, by using therescaling technique.

3.2.2.1 Rescaling

Introduce a new variable

ξ =x

δ, (3.2.16)

where δ = δ(ε). In what follows, we shall prove that in order to get an innerexpansion which matches well the outer expansion, δ should be very small, soξ is called fast variable. The first goal of this subsection is to find a correctformula of δ. Rewriting equation (3.1.1) in terms of ξ gives

ε

δ2d2U

dξ2+

1 + ε

δ

dU

dξ+ U = 0. (3.2.17)

To investigate the relation of the coefficients of (3.2.17), i.e.

ε

δ2,

1 + ε

δ, 1,

there are five cases which should be taken into account. Note that

1 + ε

δ∼ 1

δ

since ε << 1.

Case i) δ >> 1. Recalling that ε << 1, one has

ε

δ2<<

1

δ2<<

ε

δ<< 1.

Thus equation (3.2.17) becomes

ε

δ2d2U

dξ2︸ ︷︷ ︸o(1)

+1 + ε

δ︸ ︷︷ ︸o(1)

dU

dξ+ U︸︷︷︸

O(1)

= 0, (3.2.18)

so U = o(1). This large δ is not a correct scale.

Case ii) δ ∼ 1. This implies ξ ∼ x, and (3.2.16) changes nothing. In thepresent case, only a regular expansion can be expected, so that is not what wewant.

Case iii) δ << 1 and εδ2>> 1

δ. From which it follows that ε >> δ. Dividing

equation (3.2.17) by εδ2

yields

d2U

dξ2+

1 + ε

εδdU

dξ︸ ︷︷ ︸o(1)

+δ2

εU︸︷︷︸

o(1)

= 0, (3.2.19)

from which it follows thatd2U

dξ2= o(1).

Thus this scale would not lead to an inner expansion either.

Case iv) δ << 1 and εδ2

∼ 1δ. We have ε ∼ δ. Multiplying equation (3.2.17) by

δ to get

ε

δ︸︷︷︸∼1

d2U

dξ2+dU

dξ+ ε

dU

dξ+ δU︸ ︷︷ ︸

o(1)

= 0, (3.2.20)

this will lead to a correct scale. We just choose the simple relation δ = ε, and(3.2.16) turns out to be ξ = x

ε.

Case v) δ << 1 and εδ2<< 1

δ, which implies ε << δ. Multiplying equation

(3.2.17) by δ we obtain

ε

δ︸︷︷︸o(1)

d2U

dξ2+ (1 + ε)︸ ︷︷ ︸

∼1

dU

dξ+ δU︸︷︷︸

o(1)

= 0, (3.2.21)

which implies that dUdξ

= o(1). This case is not what we want.

Now we turn back to construction of inner expansions. From rescaling wecan define

ξ =x

ε, (3.2.22)

and an ansatz as follows

uε(x) = U0(ξ) + εU1(ξ) + ε2U2(ξ) + · · · . (3.2.23)

It is easy to compute that for i = 0, 1, 2, · · · ,

dUi(ξ)

dx=

1

ε

dUi(ξ)

dξ,d2Ui(ξ)

dx2=

1

ε2d2Ui(ξ)

dξ2.

Then invoking equation (3.1.1) we arrive at

ε−1 : U ′′0 + U ′

0 = 0, (3.2.24)

ε0 : U ′′1 + U ′

1 + U ′0 + U0 = 0, (3.2.25)

ε1 : U ′′2 + U ′

2 + U ′1 + U1 = 0. (3.2.26)

From which we obtain the general solutions

U0(ξ) = C01e−ξ + C02, (3.2.27)

U1(ξ) = C11e−ξ + C12 − C02ξ, (3.2.28)

U2(ξ) = C21e−ξ + C22 +

C02

2ξ2 − C12ξ. (3.2.29)

Here Cij with i = 0, 1, 2; j = 1, 2 are constants. Next step is to determinethese constants. To this end, we use the condition at x = 0 which implies thatξ = 0 too, to conclude that U0(0) = 0, U1(0) = 0 and U2(0) = 0, thus

Ci1 = −Ci2 =: Ai, (3.2.30)

for i = 0, 1, 2. Hence, (3.2.27) – (3.2.29) are reduced to

U0(ξ) = A0(e−ξ − 1), (3.2.31)

U1(ξ) = A1(e−ξ − 1) + A0ξ, (3.2.32)

U2(ξ) = A2(e−ξ − 1)− A0

2ξ2 + A1ξ. (3.2.33)

Therefore we still need to find constants Ai. For this purpose we need mat-ching conditions. An inner region is near the boundary layer and is usuallyvery thin, is of O(ε) in the present problem, while an outer region is far fromthe boundary layer. Thus there is an intermediate (or, matching, overlapping)region between them, the scale of the distance of this region to the boundarylayer is of O(εα) where α ∈ (0, 1). Inner and outer expansions are valid (bythis word we mean that an expansion satisfies well the associated equation)respectively, over inner and outer regions. Roughly speaking, matching condi-tions are the conditions which are given over the intermediate region so thatthe outer and inner expansions coincide there. The task of the next subsectionis to find such conditions.

3.2.3 Matching conditions

Now we expect reasonably that the inner expansions coincide with the outerones in the intermediate region, and write

U0(ξ) + εU1(ξ) + ε2U2(ξ) = u0(x) + εu1(x) + ε2u2(x) +O(ε3).

To derive the matching conditions, we shall employ two main methods.

3.2.3.1 Matching by expansions

(Relation with the intermediate variable method? ) Following Fife [14], werewrite x = εξ and expand the right hand side in terms of ξ. We then obtainthe matching conditions

U0(ξ) ∼ u0(0) = e, (3.2.34)

U1(ξ) ∼ u′0(0)ξ + u1(0) = −eξ, (3.2.35)

U2(ξ) ∼ 1

2u′′0(0)ξ

2 + u′1(0)ξ + u2(0) =e

2ξ2, (3.2.36)

for ξ → ∞. From (3.2.31) it follows that

U0(ξ) → A0,

as ξ → ∞. Combination with (3.2.34) yields

A0 = −e. (3.2.37)

Hence, (3.2.31) – (3.2.33) are now

U0(ξ) = −e(e−ξ − 1), (3.2.38)

U1(ξ) = A1(e−ξ − 1)− eξ, (3.2.39)

U2(ξ) = A2(e−ξ − 1) +

e

2ξ2 + A1ξ. (3.2.40)

So the leading term of inner expansion is obtained. Comparing (3.2.35) with(3.2.39) for large ξ we have

A1 = 0. (3.2.41)

In a similar manner, from (3.2.36) with (3.2.40) one gets

A2 = 0. (3.2.42)

Therefore, the first three term of the inner expansion are determined, whichread

U0(ξ) = e(1− e−ξ), (3.2.43)

U1(ξ) = −eξ, (3.2.44)

U2(ξ) =e

2ξ2. (3.2.45)

Using these functions, we define approximations up to i+ 1-terms (i = 0, 1, 2)as follows

Iε0(ξ) = e(1− e−ξ), (3.2.46)

Iε1(ξ) = e(1− e−ξ)− εeξ, (3.2.47)

Iε2(ξ) = e(1− e−ξ)− εeξ + ε2e

2ξ2. (3.2.48)

3.2.3.2 Van Dyke’s rule for matching

Matching with an intermediate variable can be tiresome. The following VanDyke’s rule [33] for matching usually works and is more convenient.

For a function f , we have corresponding inner and outer expansions whichare denoted respectively, f =

∑n ε

nfn(x) and f =∑

n εngn(ξ). We define

Definition. Let P,Q be non-negative integers.

EPf = outer limit(x fixed ε ↓ 0) retaining P + 1 terms of an outer expansion

=P∑

n=0

εnfn(x), (3.2.49)

and

HQf = inner limit(ξ fixed ε ↓ 0) retaining Q+ 1 terms of an inner expansion

=

Q∑n=0

εngn(ξ), (3.2.50)

Then the Van Dyke matching rule can be stated as

EPHQf = HQEPf.

Example. Let P = Q = 0. For our problem in this section, we define f = uε,and H0g := A0(e

−ξ − 1), E0f := e1−x. Then

E0H0g = E0A0(e−ξ − 1)

= E0A0(e−x/ε − 1)

= −A0. (3.2.51)

and

H0E0f = H0e1−x= H0e1−εξ= e. (3.2.52)

By the Van Dyke rule, (3.2.51) must coincide with (3.2.52), and we obtain

A0 = −e,

which is (3.2.37). We can also derive the matching conditions of higher order.

3.2.4 Matched asymptotic expansions

In this subsection we shall make use of the inner and outer expansions toconstruct approximations. Also we do it in two ways.

i) The first method: Adding inner and outer expansions, then subtractingthe common part, we obtain

U ε0 (x) = e1−x + e(1− e−ξ)− e = e(e−x − e−

xε )

U ε1 (x) = e(e−x − e−

xε )− ex− (−eεξ) = e(e−x − e−

xε )

U ε2 (x) = e(e−x − e−

xε ) +

1

2ex2 − 1

2e(εξ)2 = e(e−x − e−

xε ). (3.2.53)

From which one asserts that

U ε0 (x) = U ε

1 (x) = U ε2 (x).

The more terms we take, but the accuracy does not increase! This is differentfrom what we had for algebraic equations.

ii) The second method: Employing the cut-off function defined in previoussubsection. Then we get

U εi (x) = (1− χε(x))O

εi (x) + χε(x)I

εi (ξ). (3.2.54)

Here, i = 0, 1, 2.

3.3 Examples

The following examples will help us to understand the method of asymptoticexpansions.

Example 1. In a given singular perturbation problem, more than one boun-dary layer can occur. This is exemplified by

εd2u

dx2− u = A, in D. (3.3.1)

u(0) = α, u(1) = β. (3.3.2)

Here A = 0, β = 0.

Example 2. A problem can be singular although ε does not multiply thehighest order derivative of the equation. A simple example is the following:

∂2u

∂x2− ε

∂u

∂y= 0, in D = (x, y) | 0 < x < 1, 0 < y < y0. (3.3.3)

u(x, 0; ε) = f(x), for 0 ≤ x ≤ 1, (3.3.4)

u(0, y; ε) = g1(y), u(1, y; ε) = g2(y), for 0 ≤ y ≤ y0. (3.3.5)

Here we take y0 > 0, and choose u0 satisfying ∂2u0

∂x2 = 0 as follows

u0(x, y; ε) = g1(y) + (g2(y)− g1(y))x.

However, in general, u0(x, 0; ε) = f(x), so that u0 is not an approximation ofu in D.

This can be easily understood by noting that (3.3.3) is parabolic while itbecomes elliptic if ε = 0.

Example 3. In certain perturbation problems, there exists a uniquely definedfunction u0 in D satisfying the limit equation L0u0 = f0 and all the boundaryconditions imposed on u, and yet u0 is not an approximation of u:

(x+ ε)2du

dx+ ε = 0, for 0 < x < A, (3.3.6)

u(0; ε) = 1. (3.3.7)

We have the exact solution to problem (3.3.6) – (3.3.7):

u = u(x; ε) =ε

x+ ε,

and

L0 = x2d

dx.

The function u0 = 1 satisfies L0u0 = 0 and also the boundary conditions. But

limε→0

u(x; ε) =

0 if x = 0,1 if x = 0,

(3.3.8)

and

maxD

|u− u0| =A

A+ ε→ 0

as ε→ 0.

Example 4. Some operators Lε cannot be decomposed into an “unperturbedε-independent part” and “a perturbation”.

du

dx− ε exp(−(u− 1)/ε) = 0, for D = 0 < x < A, A > 0, (3.3.9)

u(0; ε) = 1− α, α > 0. (3.3.10)

Note that

limε→0

maxD

(ε exp(−(g − 1)/ε))

= 0

if and only if g > 1 for x ∈ D. Thus we do not have the decomposition withL0 =

ddx. Moreover, from the exact

u(x; ε) = 1 + ε log(x+ exp(−α/ε)),we assert easily that none of the “usually” successful methods produces anapproximation of u in D.

Chapter 4

Partial differential equations

4.1 Regular problem

Let Ω be an open bounded domain in Rn with smooth boundary ∂Ω, heren ∈ N. Consider

∆u+ εu = f0 + εf1, (4.1.1)

u|∂Ω = 0. (4.1.2)

We shall prove this is a regular problem.

4.2 Conservation laws and vanishing viscosity

method

In this section we will study the inviscid limit of scalar conservation laws withviscosity.

ut + (F (u))x = νuxx, (4.2.1)

u|t=0 = u0. (4.2.2)

The associated inviscid problem is

ut + (F (u))x = 0, (4.2.3)

u|t=0 = u0. (4.2.4)

It is a basic question that does the solution of (4.2.1) – (4.2.2) converge tothat of (4.2.3) – (4.2.4)? This is the main problem for the method of vanishingviscosity.

In this section we are going to prove that the answer to this question ispositive under some suitable assumptions. We shall make use of the methodof matched asymptotic expansions and L2-energy method.

29

4.2.1 Construction of approximate solutions

4.2.1.1 Outer and inner expansions

4.2.1.2 Matching conditions and approximations

4.2.2 Convergence

Chapter 5

An application to optimalcontrol theory

5.1 Introduction

Optimal control for hyperbolic conservation laws requires a considerable analy-tical effort and computationally expansive in practice, is thus a difficult topic.Some methods have been developed in the last years to reduce the computatio-nal cost and to render this type of problems affordable. In particular, recentlythe authors of [11] have developed an alternating descent method that takesinto account the possible shock discontinuities, for the optimal control of theinviscid Burgers equation in one space dimension. Further in [12] the vanishingviscosity method is employed to study this alternating descent method for theBurgers equation, with the aid of the Hopf-Cole formula which can be foundin [23, 36], for instance. Most results in [12] are formal.

In the present chapter we will revisit this alternating descent method inthe context of one dimensional viscous scalar conservation laws with a generalnonlinearity. The vanishing viscosity method and the method of matchedasymptotic expansions will be applied to study this optimal control problemand justify rigorously the results.

To be more precise, we state the optimal problem as follows. For a givenT > 0, we study the following inviscid problem

ut + (F (u))x = 0, in R× (0, T ); (5.1.1)

u(x, 0) = uI(x), x ∈ R. (5.1.2)

Here, F : R → R, u 7→ F (u) is a smooth function, and f denotes its derivativein the following context. The case that F (u) = u2

2is studied in e.g. [11, 12].

Given a target uD ∈ L2(R) we consider the cost functional to be minimized

31

J : L1(R) → R, defined by

J(uI) =

∫R

∣∣u(x, T )− uD(x)∣∣2 dx,

where u(x, t) is the unique entropy solution to problem (5.1.1) – (5.1.2).We also introduce the set of admissible initial data Uad ⊂ L1(R), that we

shall define later in order to guarantee the existence of the following optimi-zation problem:

Find uI,min ∈ Uad such that

J(uI,min) = minuI∈Uad

J(uI).

This is one of the model optimization problems that is often addressed in thecontext of optimal aerodynamic design, the so-called inverse design problem,see e.g. [18].

The existence of minimizers has been proved in [11]. From a practicalpoint of view it is however more important to be able to develop efficientalgorithms for computing accurate approximations of discrete minimizers. Themost efficient methods to approximate minimizers are the gradient methods.

But for large complex systems, as Euler equations in higher dimensions,the existing most efficient numerical schemes (upwind, Godunov, etc.) are notdifferentiable. In this case, the gradient of the functional is not well definedand there is not a natural and systematic way to compute its variations. Dueto this difficulty, it would be natural to explore the possible use of non-smoothoptimization techniques. The following two approaches have been developed:The first one is based on automatic differentiation, and the second one isthe so-called continuous method consisting of two steps as follows: One firstlinearizes the continuous system (5.1.1) to obtain a descent direction of thecontinuous functional J , then takes a numerical approximation of this descentdirection with the discrete values provided by the numerical scheme. Howeverthis continuous method has to face another major drawback when solutionsdevelop shock discontinuities, as it is the case in the context of the hyperbolicconservation laws like (5.1.1) we are considering here.

The formal differentiation of the continuous states equation (5.1.1) yields

∂t(δu) + ∂x(f(u)δu) = 0, in R× (0, T ); (5.1.3)

But this is only justified when the state u on which the variations are beingcomputed, is smooth enough. In particular, it is not justified when the solu-tions are discontinuous since singular terms may appear on the linearizationover the shock location. Accordingly in optimal control applications we also

need to take into account the sensitivity for the shock location (which hasbeen studied by many authors, see, e.g. [9, 19, 32]). Roughly speaking, themain conclusion of that analysis is that the classical linearized system for thevariation of the solutions must be complemented with some new equations forthe sensitivity of the shock position.

To overcome this difficulty, we naturally think of another way, namely, thevanishing viscosity method (as in [12], in which an optimal control problem forthe Burgers equation is studied) and add an artificial viscosity term to smooththe state equation. Equations (5.1.1) with smoothed initial datum then turnsout to be

ut + (F (u))x = νuxx, in R× (0, T ), (5.1.4)

u|t=0 = gε. (5.1.5)

Note that the Cauchy problem (5.1.4) – (5.1.5) is of parabolic type, thus fromthe standard theory of parabolic equations (see, for instance, Ladyzenskayaet al [27]) we have that the solution uν,ε of this problem is smooth. So thelinearized one of eq. (5.1.4) can be derived easily, which reads

(δu)t + (f(u)δu)x = ν(δu)xx, in R× (0, T ), (5.1.6)

δu|t=0 = hε. (5.1.7)

Here ν, ε are positive constants, δu denotes the variation of u. The initialdata gε, hε will be chosen suitably in Section 3, so that the perturbations ofinitial data and shock position are taken into account, this renders us that wecan select the alternating descent directions in the case of viscous conservationlaws.

To solve the optimal control problem, we also need the following adjointproblem, which reads

−pt − f(u)px = 0, in R× (0, T ); (5.1.8)

p(x, T ) = pT (x), x ∈ R, (5.1.9)

here, pT (x) = u(x, T ) − uD(x). And we smooth equation (5.1.8) and initialdata as follows

−pt − f(u)px = νpxx, in R× (0, T ); (5.1.10)

p(x, T ) = pTε (x), x ∈ R, (5.1.11)

Since solutions u = u(x, t; ν, ε), δu = δu(x, t; ν, ε) are smooth, shocks va-nish, instead quasi-shock regions are formed. Natural questions are arising as

follows: 1) How should ν, ε go to zero, more precisely, can ν, ε go to zero inde-pendently? Which one goes to zero faster, or the same? What happen if thetwo parameters ν, ε → 0? 2) What are the limit of equations (5.1.10), (5.1.6)and (5.1.4) respectively? 3) To solve the optimal control problem correctly,the states of system (5.1.3) should be understood as a pair (δu, δφ), where δφis the variation of shock position. As ν, ε → 0, is there an equation for δφwhich determines the evolution of δφ and complements equation (5.1.3)?

To answer these questions, we shall make use of the method of matchedasymptotic expansions. Our main results are: the parameters ν, ε must satisfy

ε = σν,

where σ is a given positive constant. This means that ν, ε must go to zero atthe same order, but speeds may be different. We write ε

ν= σ. Then we see

that if σ > 1, ν goes to zero faster than ε, and vice versa.We now fix ε which is assumed to be very small. As σ → ∞, namely

ν → 0, the equation of variation of shock position differs from the one derivedby Bressan and Marson [9], etc., by a term which converges to zero as σ tendsto infinity, however may be very large if σ is small enough. Thus we concludethat

1) The equation derived by Bressan and Marson is suitable for developingthe numerical scheme when σ is sufficiently large. In this case, the perturbationof initial data plays a dominant role and the effect due to the artificial viscositycan be omitted;

2) However, if σ is small, then the effect of viscosity must be taken intoaccount while the perturbation of initial data can be neglected, and a correctorshould be added.

We shall prove that the solutions to problem (5.1.4) – (5.1.5), and problem(5.1.8) and (5.1.11) converge, respectively, to the entropy solution and thereversible solution of the corresponding inviscid problems, while the solutionto problem (5.1.6) – (5.1.7) converges to the one that solves (5.1.3) in thesub-regions away from the shock, and is complemented by an equation whichgoverns the evolution of the variation of shock position.

Furthermore, using the method of asymptotic expansions we also clarifysome formal expansions used frequently in the theory of optimal control, thatthey are valid only away from shock and when some parameter is not too small.For example, for solution uν to problem (5.1.4) – (5.1.5) we normally expandit as

uν = u+ νδuν +O(ν2), (5.1.12)

where u is usually believed to be the entropy solution to problem (5.1.1) –(5.1.2), and δu is the variation of uν , the solution to (5.1.6) – (5.1.7). Ho-wever (5.1.12) is not correct near shock provided that δuν(x, 0) is bounded,

for instance. From this assumption, we have δuν is continuous and uniformlybounded by the theory of parabolic equations, moreover uν is continuous too,thus it follows from (5.1.12) that u should be continuous too, but u is normallydiscontinuous. Therefore, we should understand (5.1.12) as a multi-scale ex-pansion, and assume that u = u(x, x/ν, t), δu = δu(x, x/ν, t). We shall obtainsuch an expansion by the method of matched asymptotic expansions.

The new features to the method of asymptotic expansions in this chapeterare mainly as follows: Firstly, our expansions for uν,ε and δuν,ε are differentfrom the standard ones due to the fact that equations (5.1.6) and (5.1.4) are notindependent, (5.1.6) is the variation of (5.1.4), so when constructing asympto-tic expansions we should take this fact into account and find some compatibleconditions for the asymptotic expansions of uν,ε and δuν,ε. Secondly, we derivethe equation of variation of shock location from the outer expansions, but notfrom the inner expansions as usual, see, e.g. [14]. Our approach is based upona key observation that outer expansions converge to their values at the shockand the quasi-shock region vanishes as ν → 0.

We need to introduce some

Notations: For any t > 0, we define Qt = R × (0, t). C(t), Ca, · · · denote,respectively, constants depending on t, a, · · · , and C is a universal constant inSub-section 4.2.

For a function f = f(r, t) where r = rν(x, t; ν): ft denotes the partialderivative with respect to t while (f)t = ft + frrt, and so on.

Let X be a Banach space endowed with a norm ∥ · ∥X , and f : [0, T ] → X.For any fixed t the X-norm of f is denoted by ∥f(t)∥X , when X = L2(R), wewrite ∥f(t)∥ = ∥f(t)∥X , sometimes the argument t is omitted.

Landau symbolsO(1) and o(1). A quantity f(x, t; ν) = o(1) means ∥f∥L∞(Qt) →0 as ν → 0, and g(t; ξ) = o(1) implies that ∥g∥L∞(0,t) → 0 as ξ → ∞. Andf(x, t; ν) = O(1) means ∥f∥L∞(Qt) ≤ C uniformly for ν ∈ (0, 1]. We alsouse the standard notations: BV (R) (BV loc(R)), Lip(R) (Liploc(R)), are thespaces of the functions of (locally) bounded variations, the (locally) Lipschitzcontinuous functions in R, respectively.

The remaining parts of this chapter are as follows: In Section 2, we collectsome preliminaries and explain furthermore the motivation of this chapter. InSection 3, employing the method of matched asymptotic expansions and ta-king into account the infinitesimal perturbations of the initial datum and theinfinitesimal translations of the shock position, we shall construct, the innerand outer expansions, and obtain, by a suitable combination of the two expan-sions, the approximate solutions to problems (5.1.4)–(5.1.5), (5.1.6)–(5.1.7),and (5.1.10)–(5.1.11). Also the equations for the shock and its variation will

be derived. In Section 4 we shall prove the approximate solutions satisfy thecorresponding equations asymptotically, and converge, respectively, to thoseof the inviscid problems in a suitable sense. Finally we discuss the alternatingdescent method in the context of viscous conservation laws in Section 5, wherethe convergence results will be used.

5.2 Sensitivity analysis: the inviscid case

A solution to a hyperbolic equation may become singular after certain time,even the initial datum is smooth. Therefore in practical applications it ismore interesting to consider optimal control problems in the case that shocksappear. We shall study the optimal control problem for the inviscid equationin the present of shocks, and we focus on the particular case that solutionshave a single shock, however the analysis can be extended to consider moregeneral one-dimensional systems of conservation laws with a finite number ofnon-interacting shocks, see [8].

To develop the alternating descent method for the optimal control problemin presence of shocks, we need to investigate the sensitivity of the states ofthe system, with respect to perturbations of the initial datum and infinite-simal translations of the shock position. we shall see that equations (5.1.3)and (5.1.8) will become much more complicated. This section is devoted tointroducing some basic tools needed in the sensivity analysis.

5.2.1 Linearization of the inviscid equation

Let us firstly introduce the following hypothesis:

(H) Assume that u is a weak entropy solution to (5.1.1)–(5.1.2) with a dis-continuity along a regular curve Σ = (φ(t), t) | t ∈ [0, T ) which is Lipschitzcontinuous outside Σ. The Rankine-Hugoniot condition is satisfied on Σ:

φ′(t)[u]φ(t) = [F (u)]φ(t) .

Hereafter, we denote the jump at x = φ(t) of a piecewise smooth functionf = f(x, t) by [f ]φ(t) = f(φ(t) + 0, t)− f(φ(t)− 0, t), for any fixed t.

Note that Σ divides QT = R× (0, T ) into two subdomains Q− and Q+, tothe left and to the right of Σ respectively. see Figure 1.

As analyzed in [11], to deal correctly with optimal control and design pro-blems, the state of the system should be viewed as that constituted by the pair(u, φ) consisting of the solution of (5.1.1) and the shock position φ. The pair

t = 0

t = T

t

φ(T )

Q+Q−

Σ

φ0 x

Figure 5.1: Subdomains Q− and Q+.

(u, φ) satisfies

ut + (F (u))x = 0, in Q+ ∪Q−, (5.2.1)

φ′(t)[u]φ(t) = [F (u)]φ(t) , t ∈ (0, T ), (5.2.2)

φ(0) = φI , (5.2.3)

u(x, 0) = uI(x), x ∈ x < φI ∪ x > φI. (5.2.4)

We also need to analyze the sensitivity of (u, φ) with respect to perturba-tions of the initial datum, especially, with respect to δuI and δφI which arevariations of the initial profile uI and of the shock position φI , respectively.To be precise, we first need to introduce the functional framework based onthe generalized tangent vectors introduced in [8].

Definition 2.1 Let v : R → R be a piecewise Lipschitz continuous func-tion with a single discontinuity at y ∈ R. We define Σv as the family of allcontinuous paths γ : [0, ε0] → L1(R) with

(1) γ(0) = v and ε0 > 0 possibly depending on γ.

(2) For any ε ∈ [0, ε0] the functions uε = γ(ε) are piecewise Lipschitz witha single discontinuity at x = yε depending continuously on ε and there existsa constant L independent of ε ∈ [0, ε0] such that

|vε(x)− vε(x′)| ≤ L|x− x′|,

whenever yε ∈ [x, x′].

Furthermore, we define the set Tv of generalized tangent vectors of v as the

space of (δv, δy) ∈ L1 ×R for which the path γ(δv,δy) given by

γ(δv,δy)(ε) =

v + εδv + [v]yχ[y+εδy,y] if δy < 0,

v + εδv − [v]yχ[y,y+εδy] if δy > 0,

satisfies γ(δv,δy) ∈ Σv.Finally, we define the equivalence relation ∼ defined on Σv by

γ ∼ γ′ if and only if limε→0

∥γ(ε)− γ′(ε)∥L1

ε= 0,

and we say that a path γ ∈ Σv generates the generalized tangent vector(δv, δy) ∈ Tv if γ is equivalent to γ(δv,δy).

Remark 2.1. The path γ(δv,δy) ∈ Σv represents, at first order, the varia-tion of a function v by adding a perturbation function εδv and by shifting thediscontinuity by εδy.

Note that, for a given v (piecewise Lipschitz continuous function with asingle discontinuity at y ∈ R) the associated generalized tangent vectors (δv, δy) ∈Tv are those pairs for which δv is Lipschitz continuous with a single disconti-nuity at x = y.

Now we assume that the initial datum uI is Lipschitz continuous to bothsides of a single discontinuity located at x = φI , and consider a generalizedtangent vector (δuI , δφI) ∈ L1(R) × R. Let uI,ε ∈ ΣuI be a path whichgenerates (δuI , δφI). For sufficiently small ε the solution uε of problem (5.2.1)– (5.2.4) is Lipschitz continuous with a single discontinuity at x = φε(t), forall t ∈ [0, T ]. Thus uε generates the generalized tangent vector (δv, δφ) ∈L1(R)×R. Then it is proved in [9] that (δu, δφ) satisfies the linearized system

(δu)t + (f(u)δu)x = 0, in Q+ ∪Q−; (5.2.5)

(δφ)′(t)[u]φ(t) + δφ(t)(φ′(t)[ux]φ(t) − [f(u)ux]φ(t)

)= [f(u)δu]φ(t) − φ′(t)[δu]φ(t), t ∈ (0, T ), (5.2.6)

δφ(0) = δφI , (5.2.7)

δu(x, 0) = δuI(x), x ∈ x < φI ∪ x > φI. (5.2.8)

Remark 2.2. In this way, we can obtain formally the expansion:

(uε, φε) = (u, φ) + ε(δu, δφ) +O(ε2). (5.2.9)

Unfortunately, this expansion is, in general, not true, as we explained in theintroduction. For instance, suppose that δu is bounded and uε is continuous.

From (5.2.9) we conclude that uε converges to u uniformly, whence u shouldbe continuous too. But this is not true in general. Thus we should assume(u, δu) = (u, δu)(x, x/ε, t), a multi-scale expansion.

Remark 2.3. In Section 3, we shall see that equation (5.2.6) can not be, asexpected, recovered as the viscosity ν tends to zero. Instead, it is changed to

[u]φ(t) δφ′(t) = δφ(t)

(− [ux]φ(t)φ

′(t) + [f(u)ux]φ(t)

)+(− [δu]φ(t)φ

′(t) + [f(u)δu]φ(t)

)+1

σ

([ux]φ(t) −

([w]φ(t)φ

′(t)− [f(u)w]φ(t)) ), (5.2.10)

in which a corrector (the term involving σ) is added. Here w is a function whichwill be constructed by asymptotic expansion and has limits as x → ±φ(t) fort ∈ (0, T ).

5.2.2 Sensitivity in presence of shocks

To study the sensitivity, in the presence of shocks, of J with respect to varia-tions associated with the generalized tangent vector, we define an appropriategeneralization of the Gateaux derivative.

Definition 2.2 (Ref. [8]) Let J : L1(R) → R be a functional and uI ∈ L1(R)be Lipschitz continuous with a discontinuity at x = φI an initial datum forwhich the solution of (5.1.1) satisfies hypothesis (H). J is Gateaux differentiableat uI in a generalized sense if for any generalized tangent vector (δuI , δφI) andany family uI,ε ∈ ΣuI associated to (δuI , δφI) the following limit exists,

δJ = limε→0

J(uI,ε)− J(uI)

ε.

Moreover, it depends only on (uI , φI) and (δuI , δφI), i.e. it does not dependon the particular family uI,ε which generates (δuI , δφI). The limit is the ge-neralized Gateux derivative of J in the direction (δuI , δφI).

Then we have the proposition which characterizes the generalized Gateauxderivative of J in terms of the solution to the associated adjoint system.

Proposition 2.1 Assume that uD is continuous at x = φ(T ). The Gateauxderivative of J can be written as follows:

δJ =

∫x<φI∪x>φI

p(x, 0)δuI(x)dx+ q(0)[uI ]φIδφI ,

where the adjoint state pair (p, q) satisfies the system

−∂tp− f(u)∂x p = 0, in Q− ∪Q+, (5.2.11)

[p]Σ = 0, (5.2.12)

q(t) = p(φ(t), t), t ∈ (0, T ), (5.2.13)

q′(t) = 0, t ∈ (0, T ), (5.2.14)

p(x, T ) = u(x, T )− uD(x), x ∈ x < φ(T ) ∪ x > φ(T ),(5.2.15)

q(T ) =[F (u(x, T )− uD(x))]φ(T )

[u]φ(T )

. (5.2.16)

Remark 2.4. The backward system (5.2.11)–(5.2.16) has a unique solution.We can solve it in the following way: We first define the solution q on the shockΣ from the condition q′ = 0, with the given final value q(T ). Then this deter-mines the value of p along the shock. We then can propagate this information,together with the datum of p at time T to both sides of φ(T ), by characteristics.As both systems (5.1.1) and (5.2.11) have the same characteristics, any point(x, t) ∈ R× (0, T ) is reached backwards in time by a unique characteristics linecoming either from the shock Σ or the final data at (x, T ) (see Fig. 2). Thesolution obtained in this way coincides with the reversible solutions introducedin [6].

t = 0

t = T

t

φ(T )

Σ

φ0 xt = 0

t = T

t

φ(T )

Q+Q−

Σ

φ0 xx+x− x+x−

Figure 5.2: Characteristic lines entering on a shock(left) and subdomains Q− and

Q+ (right).

In Figure 2, we have used the following notations:

x− = φ(T )− u−(φ(T ))T, x+ = φ(T )− u+(φ(T ))T,

andQ− = (x, t) ∈ R× (0, T ) such that x < φ(T )− u−(φ(T ))t,Q+ = (x, t) ∈ R× (0, T ) such that x > φ(T )− u+(φ(T ))t.

Remark 2.5. We shall construct a solution to system (5.2.11)–(5.2.16) in thefollowing manner: We approximate the initial datum (5.1.11) by functions pTnwhich satisfiespTn are locally Lipschitz continuous, uniformly bounded in BVloc(R) such that

pTn (·, T ) → P T (·) = u(·, T )− uD(·) in L1loc(R),

and

pTn (φ(T ), T ) =[F (u(x, T )− uD(x))]φ(T )

[u]φ(T )

.

Then we first take the limit of solutions pν,n of (5.1.10) with initial data pTn ,as ν → 0 to obtain the solution pn of

−∂tp− f(u)∂xp = 0, in R× (0, T ), (5.2.17)

p(x, T ) = pTn (x), in R, (5.2.18)

the so-called reversible solution. These solutions can be characterized by thefact that they take the value pn(φ(T ), T ) in the whole region occupied by thecharacteristics that meet the shock. Thus in particular they satisfy the equa-tions (5.2.12)–(5.2.14) and (5.2.16). Moreover, pn → p as n → ∞, and ptakes a constant value in the region occupied by the characteristics that meetthe shock. Note that by construction, this constant is the same for all pn inthis region. Thus this limit solution p coincides with the solution of system(5.2.11)–(5.2.16).

In this chapter, we shall apply the method of matched asymptotic expansionsto justify these convergence results.

5.2.3 The method of alternating descent directions: In-viscid case

We shall present, in this sub-section, the main ideas of the alternating descentmethod which is introduced in [11] for the inviscid Burgers equation. Theclassification of the generalized tangent vectors into two cases is motivated bythe following proposition.

Proposition 2.2 Assume that we restrict the set of paths in ΣuI to those forwhich the associated generalized tangent vectors (δuI , δφI) ∈ TuI satisfy

δφI = −∫ φI

x− δuIdx+

∫ x+

φI δuIdx

[uI ]φI

. (5.2.19)

Then, the solution (δu, δφ) of problem (5.2.5)–(5.2.8) satisfies δφ(T ) = 0and the general Gateaux derivative of J can be written as follows:

δJ =

∫x<x−∪x>x+

p(x, 0)δuI(x)dx, (5.2.20)

where the adjoint state pair p satisfies the system

−∂tp− f(u)∂x p = 0, in Q− ∪Q+, (5.2.21)

p(x, T ) = u(x, T )− uD(x), x ∈ x < φ(T ) ∪ x > φ(T ).(5.2.22)

Analogously, when considering paths in ΣuI for which the associated gene-ralized tangent vectors (δuI , δφI) ∈ TuI satisfy that δuI = 0, then δu(x, T ) = 0and the general Gateaux derivative of J in the direction (δuI , δφI) can be writ-ten as

δJ = −[F (u(x, T )− uD(x))]φ(T )

[u]φ(T )

[uI ]φIδφI . (5.2.23)

Remark 2.6. Formula (5.2.20) establishes a simplified expression for thegeneralized Gateaux derivative of J when considering directions (δuI , δφI) thatdo not move the shock position at t = T . These directions are characterized byformula (5.2.19) which determines the infinitesimal displacement of the shockposition δφI in terms of the variation of uI to both sides of x = δφI . Note,in particular, that to any value δuI to both sides of the jump φI , a uniqueinfinitesimal translation δφI corresponds of the initial shock position that doesnot move the shock at t = T . We see that formula (5.2.20) holds even if uD isdiscontinuous at x = φ(T ), since we are dealing with a subspace of generalizedtangent vectors satisfying δφ(T ) = 0 and the Gateaux derivative of J , reducedto this subspace, is well defined.

Note also that system (5.2.21)–(5.2.22) does not allow to determine the func-tion p outside the region Q− ∪ Q+, i.e. in the region under the influence ofthe shock by the characteristic lines emanating from it. However, the value ofp in this region is not required to evaluate the generalized Gateaux derivativein (5.2.20). Analogously, formula (5.2.23) provides a simplified expression ofthe generalized Gateaux derivative of J when considering directions (δuI , δφI)

that uniquely move the shock position at t = T and which correspond to purelytranslating the shock.

Note that the results in Proposition 2.2 suggest the following decompositionof the set of generalized tangent vectors:

TuI = T 1uI ⊕ T 2

uI , (5.2.24)

where T 1uI contains those (δuI , δφI) for which identity (5.2.19) holds, and T 1

uI

the ones for which δuI = 0. Thus this provides two classes of descent directionsfor J at uI . In principle they are not optimal in the sense that they are notthe steepest descent directions but they both have three important properties:

(1) They are both descent directions.(2) They allow to split the design of the profile and the shock location.(3) They are true generalized gradients and therefore keep the structure of

the data without increasing its complexity.When considering generalized tangent vectors belonging to T 1

uI we canchoose as descent direction,

δuI =

−p(x, 0) if x < x−,

− limx→x−, x<x− p(x, 0) if x− < x < φI ,

− limx→x+, x>x+ p(x, 0) if φI < x < x+,

−p(x, 0) if x+ < x,

(5.2.25)

and

δφI = −∫ φI

x− p(x, 0)dx+∫ x+

φI p(x, 0)dx

[u]φI

, (5.2.26)

while for T 2uI a good choice is:

δuI = 0, δφI = [F (u(x, T )− uD(x))]φ(T )

[u(·, T )]φ(T )

[uI ]φI

. (5.2.27)

In (5.2.26) the value of δφI in the interval (x−, x+) does not affect the gene-ralized Gateaux derivative in (5.2.20) under the condition that δφI is chosenexactly as indicated (otherwise the shock would move and this would producean extra term on the derivative of the functional J). We have chosen the sim-plest constant value that preserves the Lipschitz continuity of δuI at x = x−

and x = x+, but not necessarily at x = δφI . Other choices would also providedescent directions for J at uI , but would yield the same Gateaux derivativeaccording to (5.2.20).

This allows us to define a strategy to obtain descent directions for J at uI

in TuI .

Based upon these studies, an alternating descent method for the optimalcontrol of the Burgers equation can thus be developed, by applying in eachstep of the descent, the following two sub-steps:

1. Use generalized tangent vectors that move the shock to search its optimalplacement.

2. Use generalized tangent vectors to modify the value of the solution at timet = T to both sides of the discontinuity, leaving the shock location unchanged.

For more details, we refer the reader to [11], and a number of numericalexperiments show that this method is much more robust and efficient than theusual ones.

In this chapter, motivated by the strategy for obtaining descent directionsfor J in the inviscid case, we will carry out, in Section 5, this procedure forviscous conservation laws provided that the parameter ν is sufficiently small.

We now turn to discuss the existence of minimizers of the functional J ,depending on a small parameter that comes from the solutions to the viscousproblems, over suitable admissible set and their limit as the small parametertends to zero. Let us now introduce the set of admissible initial data Uad ⊂L1(R), which is

Uad = f ∈ L∞(R) | supp(f) ⊂ K, ∥f∥L∞(R)) ≤ C, (5.2.28)

where K ⊂ R is a bounded interval and C > 0 a constant. We shall see laterthis choice guarantees the existence of minimizers for the following optimiza-tion problem:Find uI,min ∈ Uad such that

J(uI,min) = minuI∈Uad

J(uI). (5.2.29)

To make the dependence on the viscosity parameter ν more explicit the func-tional J will be denoted by Jν , although its definition is the same as that ofJ . Similarly we now consider the same minimization problem for the viscousmodel (5.1.4):Find uI,min ∈ Uad such that

Jν(uI,min) = min

uI∈Uad

Jν(uI). (5.2.30)

For the existence of minimizers of the functionals J and Jν , and the conclu-sion that the minimizers of the viscous problem (ν > 0) converge to a minimizerof the inviscid problem as the viscosity goes to zero, we refer to a recent work[12] for the vanishing viscosity method for the inviscid Burgers equation. Inthat paper, the following theorems are proved.

Theorem 5.2.1 (Existence of Minimizers) Assume that Uad is defined in (5.2.28)and uD ∈ L2(R). Then the minimization problem (5.2.29) and (5.2.30) haveat least one minimizer uI,min ∈ Uad.

Theorem 5.2.2 Any accumulation point as ν → 0 of uI,minν , the minimizers

of (5.2.30), with respect to the weak topology in L2, is a minimizer of thecontinuous problem (5.2.29).

Note that for any positive ν solutions δu, p of equations (5.1.6) and (5.1.10)are smooth, thus the Gateaux derivative of the functional J is as follows

δJ = < δJ(uI), δuI > =

∫Rp(x, 0)δuI(x)dx, (5.2.31)

where the adjoint state p = pν is the solution to (5.1.10) with initial datump(x, T ) = u(x, T )− uD.

Unlike in the inviscid one, the adjoint state now has only one component.Indeed, there is no adjoint shock variable since the state does not presentshocks. Similarly, the derivative of J has also only one term. According tothis, the straightforward application of a gradient method for the optimizationof J would lead, in each step of the iteration, to make use of the variationpointing in the direction

δuI = −p(x, 0), (5.2.32)

where p = pν is the solution to the viscous dual problem. So the alternatingmethod is considerably simplified. But, when processing this way, we wouldnot be exploiting the possibilities that the alternate descent method provides.Therefore, we take into account the effects of possible infinitesimal perturba-tions of initial datum and also infinitesimal translations, and use variations ofthe form

uIε(x) = uIε(x+ εδφI) + εδuIε(x), (5.2.33)

where, φI stands for a reference point on the profile of uI , not necessary apoint of discontinuity. When uI has a point of discontinuity, φI could be itslocation and δφI an infinitesimal variation of it. However φI could also beanother singular point on the profile of uI , as, for instance, an extremal pointor a point where the gradient of uI is large, namely a smeared discontinuity.By a Taylor expansion, (5.2.33) can be rewritten in the following form

uIε(x) = uI(x) + ε(δφIuIx(x) + δuI(x)

)+O(ε2). (5.2.34)

This indicates that the result of these combined variations (δuI , δφI) is equi-valent to a classical variation in the direction of δφIuIx+ δu

I . We find that theeffect of a small δφI can be amplified by a large gradient δuI .

As we will see in the next section, (5.2.33) and (5.2.34) give us some hintson how to construct outer expansions that include the effects of possible infi-nitesimal perturbations of initial datum and infinitesimal translations.

5.3 Matched asymptotic expansions and ap-

proximate solutions

In this section, we are going to apply the method of matched asymptoticexpansions to construct inner and outer expansions. Then we get the ap-proximate solutions by a suitable combination of them. Firstly, we derive theouter expansions. For the purpose of sensitivity analysis of the states of thesystem, we add infinitesimal perturbations to the initial datum and infinite-simal translations to the shock position, and make use of Taylor expansions.These operations make it possible to derive the equations for the shock andits variation.

Let v = δu, ψ = δφ for simplicity. We consider asymptotic expansions ofsolutions to the following systems.

ut + (F (u))x = νuxx, in R× (0, T ), (5.3.1)

u|t=0 = gε, (5.3.2)

vt + (f(u)v)x = νvxx, in R× (0, T ), (5.3.3)

v|t=0 = hε, (5.3.4)

−pt − f(u)px = νpxx, in R× (0, T ), (5.3.5)

p|t=T = pTn,ε. (5.3.6)

Equation (5.3.1) is the usual conservation law with viscosity, (5.3.3) is itslinearized equation, and (5.3.5) is the dual equation of (5.3.3).

We make the following assumptions:

Assumptions. A1)

The function F is smooth. (5.3.7)

A2) Let n ∈ N. g0, h0 ∈ L1(R) ∩ L∞(R), g0x ∈ L1(R) and pTn ∈ L1(R) ∩Liploc(R) ∩ L∞(R). g0x, h0 have only a shock at φI , respectively. There existsmooth functions gε, hε, pTn,ε ∈ C∞

0 (R), such that

gε, hε, pTn,ε → g0, h0, pTn (5.3.8)

in L2(R) as ε → 0, respectively. Moreover, we assume that pTn is boundedsequence in BV loc(R) such that as n→ ∞,

pTn → pT in L1loc(R). (5.3.9)

A3) Assume further that Oleinik’s one-sided Lipschitz condition (OSLC) issatisfied, i.e.

(f(u(x, t))− f(u(y, t))) (x− y) ≤ α(t)(x− y)2 (5.3.10)

for almost every x, y ∈ R and t ∈ (0, T ), where α ∈ L1(0, T ).

5.3.1 Outer expansions

An outer expansion is valid outside the interfacial (or quasi-shock) region, itis a series of the following form

η = η(x, t, ν) = η0(x, t) + ν η1(x, t) + ν2 η2(x, t) + · · · , (5.3.11)

or

η = η(x, t, ν) = η0(x, t) + ν η1(x, t) + ν2 η2(x, t) + · · · , (5.3.12)

where η, η0, η0, η1, η

1, · · · depend on variables t, x (but, do not depend onfast variable rν) and will be replaced by u, u0, u

0, u1, u1, · · · , respectively.

So do the initial data h, g, pTn . Hereafter, we use · · · to denote the remainderwhich has higher order of the small parameter ν, thus can be omitted.

Step 1. Outer expansions of u. We start with the construction ofouter expansion of u. The following conventions will apply throughout thissubsection: ui, uij denote a function depending on i and i, j, respectively, wherethe i, j are non-negative integers. The k-th power of ui is denoted by (ui)k.However, if ν is a parameter, νi still stands for the i-th power of ν.

We now expand the initial data as follows

hε = h0(x) + εh1(x) + ε2h2(x) + · · · , (5.3.13)

pTn,ε = pTn0(x) + εpTn1(x) + ε2pTn2(x) + · · · , (5.3.14)

and the nonlinear terms as

F (η) = F (η0) + νf(η0)η1 + ν2(f(η0)η2 +

1

2f(η0)(η1)

2

)+ · · · ,(5.3.15)

f(η) = f(η0) + νf ′(η0)η1 + ν2(f ′(η0)η2 +

1

2f ′′(η0)(η1)

2

)+ · · · .(5.3.16)

However, the initial datum gε will be expanded in a special manner.

In order to analyze the sensitivity of the states with respect to perturbationsof the initial data and the shock position, we construct the outer expansionsin the following way: It is easy to see that the solution u to equation (5.3.1)with (5.3.13) depends on two parameters ν, ε, so we write

u = u(x, t; ν, ε),

and define

xε = x− ε(ν)ψ(t), (5.3.17)

where ε = ε(ν) is a function to be determined. Then making a transformationof variable x→ xε, we obtain a new function

u = u(xε, t; ν, ε) := u(xε + ε(ν)ψ(t), t; ν, ε) = u(x, t; ν, ε).

For the initial data, we make the same transformation of variable, i.e.(5.3.17) with t = 0 , and have

gε(x) = g0(xε) + εg1(xε) + ε2g2(xε) + · · · . (5.3.18)

Here, we defined gi(xε) = gi(xε + εψ(0)) with i = 0, 1, 2.

In order to get the outer expansion of u in the form (5.3.11), we need to doit in two steps. Firstly we expand u(xε, t; ν, ε) in terms of ε and (η, t) = (xε, t)(we now regard xε as an independent variable) as follows

u = u0 + εu1 + ε2u2 + · · · , (5.3.19)

where ui = ui(xε, t; ν), for i = 0, 1, 2, · · · , and we find, by testing differentansatze, that in order to get asymptotic expansions there must hold

ε = σν. (5.3.20)

Secondly we expand ui in terms of ν and (x, t) as follows. For i = 0, invoking(5.3.17), by Taylor expansions and (5.3.20), one has

u0(xε, t; ν) = u00(xε, t) + νu01(xε, t) + ν2u02(xε, t) + · · ·

= u00(x, t) + u00,x(x, t)εψ +1

2u00,xx(x, t)(εψ)

2 + ν(u01(x, t) + u01,x(x, t)εψ

)+ ν2u02(x, t) + · · ·

= u00(x, t) + ν(u00,x(x, t)σψ + u01(x, t)

)+ ν2

(1

2u00,xx(x, t)(σψ)

2 + u01,x(x, t)σψ + u02(x, t)

)+ · · · .(5.3.21)

In a similar manner, we obtain, for i = 1, 2, that

u1(xε, t; ν) = u10(xε, t) + νu11(xε, t) + · · ·= u10(x, t) + ν

(u10,x(x, t)σψ + u11(x, t)

)+ · · · , (5.3.22)

and

u2(xε, t; ν) = u20(xε, t) + · · · = u20(x, t) + · · · . (5.3.23)

Therefore, from (5.3.19) – (5.3.23) we get an ansatz for u as follows

u = u0 + νu1 + ν2u2, (5.3.24)

here, u0, u1 and u2 are defined by

u0 = u00, (5.3.25)

u1 = u01 + σu10, (5.3.26)

u2 = u02 + σu11 + σ2u20. (5.3.27)

Straightforward computations show that (5.3.1) can be written in terms ofui and ε as

0 = ut + (F (u))x − νuxx

= u0t + (F (u0))x − νu0xx + ε(u1t +

(f(u0)u1

)x− νu1xx

)+ ε2

(u2t + (f(u0)u2 +

f ′(u0)

2(u1)2)x − νu1xx

)+ · · · . (5.3.28)

Inserting (5.3.24) into (5.3.28), recalling ε = σν, then equating the coefficientsof νk (where k = 0, 1, 2) on both sides of the resulting equation, we thus obtain

ν0 : (u0)t + (F (u0))x = 0, (5.3.29)

ν1 : (u1)t + (f(u0)u1)x = R, (5.3.30)

ν2 : (u2)t + (f(u0)u2)x = R1, (5.3.31)

where R and R1 are functions defined by

R := u0,xx − σ(u0,xψ

′ + ((u0,x)t + (f(u0)u0,x)x)ψ)

and

R1 :=

(1

2u00,xσψ + u01

)xx

−(1

2u00,xx(σψ)

2 + u01,xσψ

)t

− σ(u10,xσψ

)t

−(f(u00)

(1

2u00,xx(σψ)

2 + u01,xσψ

)+

1

2f ′(u00)(u

00,xσψ + u01)

2

)x

−σ(f(u00)u

10,xσψ + f ′(u00)(u

00,xσψ + u01)u

10

)x+ σ(u10)xx − σ2

(1

2f ′(u00)(u

10)

2

)x

=

(1

2u0,xσψ + u1

)xx

−(1

2u0,xx(σψ)

2 + u1,xσψ

)t

−(f(u0)

(1

2u0,xx(σψ)

2 + u1,xσψ

)+

1

2f ′(u0)(u0,xσψ + u1)

2

)x

(5.3.32)

from which we see that R, R1 depend on constant σ and functions ψ, u0 andu1.

We now consider the expansion of the initial data. By using again Taylorexpansions, (5.3.18) can be rewritten as

gε(x) = g0(x) + σν (ψ(0)g′0(x) + g1(x))

+σ2ν2(1

2ψ(0)2g′′0(x) + ψ(0)g1(x) + g2(x)

)+ · · · . (5.3.33)

Thus this expansion suggests us to choose the initial data of ui (i = 0, 1, 2) asfollows

u0|t=0 = g0, (5.3.34)

u1|t=0 = g1, (5.3.35)

u2|t=0 = g2. (5.3.36)

Here, g0, g1, g2 are defined by

g0 = g0, (5.3.37)

g1 = σ (ψ(0)g′0 + δg0) , (5.3.38)

g2 = σ2

(1

2ψ(0)2g′′0 + ψ(0)g1(x) + g2(x)

). (5.3.39)

From (5.3.38) it follows that νg1 is just the second term in (5.2.34). So theexpansion of gε obtained here coincides with the one carried out formally in(5.2.34) in Section 2.

To solve the equations (5.3.29)–(5.3.31) with initial data (5.3.34)–(5.3.36),we need to get the equations for φ and ψ. So their solvability will be left tosub-section 3.3, where the interface equations will be derived.

Step 2. Outer expansions of v. Now we turn to derive the outer ex-pansions for v. It is not necessary to take into account again the effects ofinfinitesimal perturbations of the initial datum and infinitesimal translationsof the shock position, since we have had, in hand, the equations for shock andits perturbation. We can construct directly an ansatz for v, that has the sameform of (5.3.24), as follows

v = v0 + νv1 + ν2v2. (5.3.40)

Similarly, from equation (5.3.3) we then have

ν0 : (v0)t + (f(u0)v0)x = 0, (5.3.41)

ν1 : (v1)t + (f(u0)v1)x = δ1, (5.3.42)

ν2 : (v2)t + (f(u0)v2)x = δ2, (5.3.43)

and the initial data are

v0|t=0 = h0, (5.3.44)

v1|t=0 = h1, (5.3.45)

v2|t=0 = h2. (5.3.46)

Here δ1, δ2 are given by

δ1 = (v0)xx − (f ′(u0)u1v0)x ,

δ2 = (v1)xx −(f ′(u0)u2v0 +

1

2f ′′(u0)(u1)

2v0

)x

. (5.3.47)

By solving these problems (which are initial value problems of transportequations with discontinuous coefficients, for the well-posedness we refer toe.g. [6]), we then construct v0, v1, v2 which are smooth up to the shock.

Step 3. Outer expansions of p. To get the outer expansions for p werepeat again the procedure performed for v, similarly, from equation (5.3.5)we then have

ν0 : (p0)t + f(u0)(p0)x = 0, (5.3.48)

ν1 : (p1)t + f(u0)(p1)x = β1, (5.3.49)

ν2 : (p2)t + f(u0)(p2)x = β2, (5.3.50)

and the initial data are

p0|t=0 = pT0 , (5.3.51)

p1|t=0 = pT1 , (5.3.52)

p2|t=0 = pT2 . (5.3.53)

Here β1, β2 are given by

β1 = − (f ′(u0)u1(p0)x + (p0)xx) , (5.3.54)

β2 = −(f ′(u0)u1(p1)x +

(f ′(u0)u2 +

f ′′(u0)

2(u1)

2

)(p0)x + (p1)xx

).(5.3.55)

Thus we first solve p0, then insert p0 into equations p1 and p2, from theresulting linear equations we then can obtain p1 and p2.

We are now going to discuss the solvability of the problems (5.3.29) –(5.3.36). There are five unknowns, i.e. ui (i = 0, 1, 2), φ and ψ, but onlythree equations. So, to form a complete system, as the first step, we need tofind the equations that φ and ψ satisfy.

5.3.2 Derivation of the interface equations

In this sub-section we shall derive the interface equations from the outer ex-pansions of u, this derivation is based upon an important observation that thevalues of outer expansions tend to those at the shock φ(t), as the thicknessν of the quasi-shock region goes to zero. This is different from the usual waythat one derives such equations from inner expansions, see, e.g. [14, 16]. Oneadvantage of this approach is that we can overcome the difficulty/limitationcaused by the algebraically growth, as rν → ∞, which is required by the mat-ching conditions, i.e. (5.3.73), of the second term i.e. u1 of inner expansionof u, so how to define the jump (e.g. u1(∞, t) − u1(−∞, t), but u1(±∞, t)may not exsist) of the terms like u1, of the inner expansion of u? We cannotdefine it in a usual way, some restrictions must be imposed, for instance, addan assumption that ∂xu0(φ(t)± 0, t) = 0 for u1.

Assume that the limits of ui (or its derivatives) exist for any t ∈ [0, T ], fromthe left and right side of the shock, i.e.

∂kxu0(φ(t)± 0, t), k = 0, 1, and u1(φ(t)± 0, t), exist.

First of all, we derive the equation for the shock, and invoke equation(5.3.29) with initial datum (5.3.34) which is just a Cauchy problem for theunknown u0. From the standard theory it then follows that

φ′ [u0]φ(t) = [F (u0)]φ(t), (5.3.56)

which is the Rankine-Hugoniot condition.Next we make use of (5.3.30) to find the equation for the variation of the

shock position, i.e. ψ. We define, in a usual way, weak solution to problem(5.3.30) – (5.3.35), namely multiplying (5.3.30) by a test function ζ with com-pact support in R × [0, T ), and integrating it by parts. Then similar to thederivation of the Rankine-Hugoniot condition, we derive, as e.g. in Smoller[31], the equation of ψ from this weak formulation by choosing a test functionζ with compact support contained in a small neighborhood D = D1 ∪D2 of afixed point (x, t), see Figure 3. Then applying integration by parts yield that

σ[u0]φ(t)ψ′(t) + σ

( (−φ′(t) [u0,x]φ(t) + [f(u0)u0,x]φ(t)

)ψ(t)

)= φ′(t) [u1]φ(t) − [f(u0)u1]φ(t) + [u0,x]φ(t). (5.3.57)

Here we used the fact that the integral vanishes over interior of D1 and D2,and that the values of ζ are equal to zero at the boundary of D.

x

t

O

(x, t)D1

D2

Σ

D

Figure 5.3: Subdomains D1 and D2.

Dividing equation (5.3.57) by σ we obtain

[u0]φ(t)ψ′(t) +

(−φ′(t) [u0,x]φ(t) + [f(u0)u0,x]φ(t)

)ψ(t)

=1

σ

( (φ′(t) [u1]φ(t) − [f(u0)u1]φ(t)

)+ [u0,x]φ(t)

).(5.3.58)

Remark 3.1. By (5.3.26) we can rewrite the right-hand side of (5.3.58) as

1

σ

( (φ′(t) [u1]φ(t) − [f(u0)u1]φ(t)

)+ [u0,x]φ(t)

)=

(φ′(t) [u10]φ(t) − [f(u0)u

10]φ(t)

)+

1

σ

( (φ′(t) [u10]φ(t) − [f(u0)u

10]φ(t)

)+ [u0,x]φ(t)

)→ φ′(t) [u10]φ(t) − [f(u0)u

10]φ(t). (5.3.59)

Thus we see that if f(u) = u, equation (5.3.58) converges to the one derivedin [8].

We turn back to solvability of equation (5.3.58). To this end we need todetermine a priori the quantity [u1]φ(t). Invoking outer expansions of uν andvν , noting that vν is the variation of uν , we have two expansions of uν , whichare valid up to the shock,

u(x, t; ν, σν) =: uν = u0 + νu1 + · · · (5.3.60)

= u+ νvν + · · · = u+ νv0 + · · · . (5.3.61)

Here u is believed to be the entropy solution to problem (5.1.1)–(5.1.2). So itis natural to assume that u0 = u, whence u1 = v0. Thus, there must hold

limx→φ(t)

u1(x, t) = limx→φ(t)

v0(x, t) = v0(φ(t), t)

for any t ∈ [0, T ].Therefore, we can write [u1]φ(t) in terms of [v0]φ(t). From (5.3.56) we get

φ = φ(t) and insert it into (5.3.58). Hence from (5.3.58) one solves (5.3.58)and gets ψ. Furthermore, we can solve u0 from equation (5.3.29) with (5.3.34)which form a Riemann problem. Once u0 is obtained, we see that the righthand side of (5.3.30) and (5.3.31), namely R, R1 are also known quantities,thus problem (5.3.30) with (5.3.35) and problem (5.3.31) with (5.3.36) arelinear in u1, u2 respectively, and can be solved easily to get u1, u2.

Combining the results obtained in sub-sections 3.1 and 3.2, we then com-plete the construction of outer expansions of u, v and p. We also find that thefirst term v0 of the outer expansion of v(x, t; ν, ε) can be expected to solve

(v0)t + (f(u0)v0)x = 0, in Q+ ∪Q−; (5.3.62)

[u0]φ(t)ψ′(t) +

(−φ′(t) [u0,x]φ(t) + [f(u0)u0,x]φ(t)

)ψ(t)

=1

σ

( (φ′(t) [v0]φ(t) − [f(u0)v0]φ(t)

)+ [u0,x]φ(t)

),(5.3.63)

ψ(0) = δφI , (5.3.64)

v0(x, 0) = vI(x), x ∈ x < φI ∪ x > φI. (5.3.65)

Note that (5.2.6) is replaced by (5.3.63).

Remark 3.2. By the method of characteristics, we solve u1 from problem(5.3.30), (5.3.35), and write it as

u1(x, t) = u1(x(0), 0)

∫ t

0

R(x, s) · exp(∫ s

t

(f(u0(x, τ)))xdτ

)ds,

for (x, t) ∈ Q+ ∪Q−, and ψ from (5.3.58). Thus we see that ψ, u1 depend onthe parameter σ. On the other hand, under assumption that ψ depends on σ,checking the construction we find that functions u01 and u

10 are still independent

of σ. Therefore, by formula (5.3.26) we assert that

u01 = u1(x, t; σ)|σ=0 = u1(x(0), 0)

∫ t

0

u0,xx(x, s) exp

(∫ s

t

(f(u0(x, τ)))xdτ

)ds,

whence

u10 =1

σ(u1 − u01). (5.3.66)

Note that u01 is the second term of outer expansions suppose that there is noperturbation of the initial datum. It is possible that the right hand side of(5.3.66) is independent of σ.

5.3.3 Inner expansions

An inner expansion is the one that is valid inside the interfacial region, whichcan be written as a series consisting of terms in fast variable r = rν and time t

η = η(rν , t, ν) = η0(r, t) + νη1(r, t) + ν2η2(r, t) + · · · ,

here η, η0, η1, · · · will be replaced by u, u0, u1, · · · , v, v0, v1, · · · andp, p0, p1, · · · , respectively.

The nonlinear terms can be expanded as

F (η) = F (η0) + νf(η0)η1 + ν2(f(η0)η2 +

1

2f(η0)η

21

)+ · · · , (5.3.67)

f(η) = f(η0) + νf ′(η0)η1 + ν2(f ′(η0)η2 +

1

2f ′′(η0)η

21

)+ · · · . (5.3.68)

In what follows, for implicity, we use the notation

h′ =∂

∂rνh

for a function h in fast variable. Then we have

∂h

∂dν=

1

νh′,

∂2h

∂dν2 =

1

ν2h′′.

Therefore, we have

ν−1 : u′′0 + φu′0 − (F (u0))′ = 0, (5.3.69)

ν0 : u′′1 + φu′1 − (f(u0)u1)′ = −σψu′0 + u0t, (5.3.70)

ν1 : u′′2 + φu′2 − (f(u0)u2)′ = −σψu′1

(f ′(u0)

2u21

)′

+ u1t. (5.3.71)

Then we use the matching conditions which make the inner and outer expan-sions coincide asymptotically in an intermediate region, say M. Over thisregion, there holds

u0(rν , t)+νu1(rν , t)+ν2u2(rν , t) = u0(x, t)+νu1(x, t)+ν

2u2(x, t)+O(ν3), inM.

By definition, we can rewrite x = ν(rν + σψ(t)) + φ(t). Then using Taylorexpansions we can obtain the matching conditions in the following form

u0(rν , t) = u0(φ(t)± 0, t) + o(1), (5.3.72)

u1(rν , t) = u1(φ(t)± 0, t) + (rν + σψ)∂xu0(φ(t)± 0, t) + o(1),(5.3.73)

u2(rν , t) = u2(φ(t)± 0, t) + (rν + σψ)∂xu1(φ(t)± 0, t)

+(rν + σψ)2

2∂2xu0(φ(t)± 0, t) + o(1), (5.3.74)

for more details, we refer, e.g. to the book by Fife [14]. Later we will chooseo(1) = exp(−cξ2), where c is a fixed positive number.

Therefore the problems (5.3.69) – (5.3.71) with corresponding boundaryconditions (5.3.72) – (5.3.74), which are boundary value problems of ordinarydifferential equations of second order, have a unique solution, respectively,provided the orthogonality conditions are met:∫ ∞

−∞(−σψu′0 + u0t)p

∗dr = 0 and

∫ ∞

−∞

(−σψu′1

(f ′(u0)

2u21

)′

+ u1t

)p∗dr = 0(5.3.75)

where p∗ satisfied L∗(p∗) = 0, and L∗ is the adjoint operator of L which isdefined by

L(u1) = u′′1 + φu′1 − (f(u0)u1)′.

We thus obtain u0, u1, u2.

Straightforward computations yield the equations for the inner expansionof v

ν−1 : v′′0 + φv′0 − (f(u0)v0)′ = 0, (5.3.76)

ν0 : v′′1 + φv′1 − (f(u0)v1)′ = f1, (5.3.77)

ν1 : v′′2 + φv′2 − (f(u0)v2)′ = f2, (5.3.78)

where f1 and f2 are defined by

f1 = v0t − σψv′0 + (f ′(u0)u1v0)′, (5.3.79)

f2 = v1t − σψv′1 +

(f ′(u0)u1v1 +

(f ′(u0)u2 +

1

2f ′′(u0)u

21

)v0

)′

.(5.3.80)

Using the matching conditions

v0(rν , t) = v0(φ(t)± 0, t) + o(1), (5.3.81)

v1(rν , t) = v1(φ(t)± 0, t) + (rν + σψ)∂xv0(φ(t)± 0, t) + o(1),(5.3.82)

v2(rν , t) = v2(φ(t)± 0, t) + (rν + σψ)∂xv1(φ(t)± 0, t)

+(rν + σψ)2

2∂2xv0(φ(t)± 0, t) + o(1), (5.3.83)

Solving problems (5.3.76) – (5.3.78) with suitable orthogonality conditions andalso boundary conditions (5.3.81) – (5.3.83), we then obtain v0, v1, v2.

Finally we construct the inner expansion of p, the first three terms of it canbe obtained by solving the following equations

ν−1 : p′′0 − φp′0 + f(u0)p′0 = 0, (5.3.84)

ν0 : p′′1 − φp′1 + f(u0)p′1 = f3, (5.3.85)

ν1 : p′′2 − φp′2 + f(u0)p′2 = f4, (5.3.86)

where f3 and f4 are defined by

f3 = ψp′0 − p0t − f ′(u0)u1p′0, (5.3.87)

f4 = ψp′1 − p1t − f ′(u0)u1v1 −(f ′(u0)u2 +

1

2f ′′(u0)u

21

)p′0. (5.3.88)

One can easily find that equations satisfied by pi (i = 0, 1, 2), namely (5.3.84)– (5.3.86) are just the dual ones of those satisfied by vi, i.e. (5.3.76) – (5.3.78).

From the following matching conditions

p0(rν , t) = p0(φ(t)± 0, t) + o(1), (5.3.89)

p1(rν , t) = p1(φ(t)± 0, t) + (rν + σψ)∂xp0(φ(t)± 0, t) + o(1),(5.3.90)

p2(rν , t) = p2(φ(t)± 0, t) + (rν + σψ)∂xp1(φ(t)± 0, t)

+(rν + σψ)2

2∂2xp0(φ(t)± 0, t) + o(1), (5.3.91)

we solve uniquely, under suitable orthogonality conditions, equations (5.3.84)– (5.3.86) and get p0, p1, p2.

Therefore, we have constructed the inner and outer expansions up to ν2, foru, v, p respectively, which can be written as follows

U2(x, t) = u0 + νu1 + ν2u2, (5.3.92)

U2(r, t) = u0 + νu1 + ν2u2. (5.3.93)

V2(x, t) = v0 + νv1 + ν2v2, (5.3.94)

V2(r, t) = v0 + νv1 + ν2v2. (5.3.95)

P2(x, t) = p0 + νp1 + ν2p2, (5.3.96)

P2(r, t) = p0 + νp1 + ν2p2. (5.3.97)

5.3.4 Approximate solutions

In this subsection we shall use a suitable cut-off function to combine togetherthe outer and inner expansions derived in sub-sections 3.1, 3.3 and 3.4, whencethe approximate solutions will be constructed.

We define a function χ = χ(ξ) : R → R+ which is smooth such that

χ(ξ) =

1 if |ξ| ≤ 1,

0 if |ξ| ≥ 2,(5.3.98)

and 0 ≤ χ(ξ) ≤ 1 if ξ ∈ [1, 2]. And let

χν(x, t) = χ(ν1−γrν), (5.3.99)

which is easily seen from Figure 4 that

supp(χν) ⊂ [0, 2νγ], supp(χ′ν), supp(χ

′′ν) ⊂ [νγ, 2νγ].

From the expansions (5.3.92) and (5.3.93), (5.3.94) and (5.3.95) and (5.3.96)and (5.3.97), we are in a position to construct, respectively, the approximatesolutions U ν

2 , Vν2 and P ν

2 as follows

U ν2 (x, t) = χν(x, t)U2(rν , t) + (1− χν(x, t))U2(x, t), (5.3.100)

V ν2 (x, t) = χν(x, t)V2(rν , t) + (1− χν(x, t))V2(x, t), (5.3.101)

P ν2 (x, t) = χν(x, t)P2(rν , t) + (1− χν(x, t))P2(x, t). (5.3.102)

By definition, we find easily that if (x, t) is sufficiently close to the quasi-shockx = φ(t), then χν(x, t) = 1, thus the approximate solution U ν

2 is equal to innerexpansion U2, namely

U ν2 (x, t) = U2(rν , t),

νγ 2νγ0

1

−νγ−2νγ

νγ 2νγ

0−νγ−2νγ

ν−γ

−ν−γ

Figure 5.4: Typical shapes of functions χν and χ′ν .

on the other hand, suppose that if (x, t) is sufficiently away from the quasi-shock, then 1−χν(x, t) = 1 which yields that the approximate solution is justthe outer expansion, i.e.

U ν2 (x, t) = U2(x, t).

In the intermediate region, there hold 0 ≤ χν(x, t) ≤ 1 and U2(rν , t), U2(x, t)are equal asymptotically which can be seen from the matching conditions, thuswe can replace one of the two expansions by one another with a small error,to be precise, we write

U ν2 (x, t) = χν(x, t)U2(rν , t) + (1− χν(x, t)) U2(rν , t) + o(1) = U2(rν , t) + o(1),

also there holds U ν2 (x, t) = U2(x, t) + o(1). So the approximate solution is a

good combination of inner and outer expansions.In what follows we will omit arguments (x, t), (rν , t) and so on, for simplicity.

Theorem 5.3.1 Suppose that the condition (5.3.75) and assumptions (5.3.7)– (5.3.10) are satisfied and that ε = σν with σ being a positive constant.

Then the approximate solutions Uν2 , V

ν2 , P

ν2 satisfy respectively, equations

(5.3.1), (5.3.3) and (5.3.5), in the following sense

(U ν2 )t − ν(U ν

2 )xx + (F (U ν2 ))x = O(να), (5.3.103)

(V ν2 )t − ν(V ν

2 )xx + (f(U ν2 )V

ν2 )x = O(να), (5.3.104)

−(P ν2 )t − ν(P ν

2 )xx − f(U ν2 )(P

ν2 )x = O(να), (5.3.105)

where α = 3γ − 1 and γ ∈ (13, 1).

5.4 Convergence of the approximate solutions

This section is devoted to the proof of following Theorem 4.1, which consistsof two parts, one is to prove Theorem 3.1 which asserts that the equations aresatisfied asymptotically, and another is to investigate the convergence rate.

Theorem 5.4.1 Suppose that the assumptions in Theorem 3.1 are met. Letu, p be, respectively, the unique entropy solution with only one shock, the re-versible solution to problems (5.1.1) – (5.1.2) and (5.1.8) – (5.1.9), and let vbe the unique solutions to problem (5.3.62) – (5.3.65), such that∫ T

0

∫x =φ(t), t∈[0,T ]

6∑i=1

(|∂ixu(x, t)|2 + |∂ixv(x, t)|2 + |∂ixp(x, t)|2

)dxdt ≤ C.(5.4.1)

Then the solutions (uν , vν) of problems (5.3.1) – (5.3.2) and (5.3.3) – (5.3.4)converge, respectively, to (u, v) in L∞(0, T ;L2(R))×L∞(0, T ;L2(R)), and thefollowing estimates hold

sup0≤t≤T

∥uν(t)− u(t)∥+ sup0≤t≤T

∥vν(t)− v(t)∥ ≤ Cηνη. (5.4.2)

The solution pν,n of problem (5.3.5) – (5.3.6) converges to p in L∞(0, T ;L1loc(R)),

namely,

sup0≤t≤T

∥pν,n(t)− p(t)∥L1loc(R) → 0, (5.4.3)

as first ν → 0, then n → ∞. Moreover, we also have in a sub-domain thefollowing estimate

sup0≤t≤T

∥pν,n − pn∥L∞(Ωh) ≤ Cν → 0, (5.4.4)

as ν → 0. Here η is a constant defined by

η = min

3

2γ,

1 + γ

2

, where γ is the same as in Theorem 3.1, (5.4.5)

and Cη denotes a constant depending only on parameters η. The domain Ωh

is defined, for any positive constant h, by

Ωh = (x, t) ∈ QT | |x− φ(t)| > h.

Remark 4.1. Combination of (5.4.4) and a stability theorem of the reversiblesolutions (see Theorem 4.1.10 in Ref. [6])) yields that

sup0≤t≤T

∥pν,n − p∥L∞(Ωh∩[−R,R]) → 0, (5.4.6)

as ν → 0, then n → ∞, where R is any positive constant. This ensures us toselect alternating descent directions.

5.4.1 The equations are satisfied asymptotically

In this sub-section, we are going to prove that the approximate solutionsU ν2 , V

ν2 , P

ν2 satisfy asymptotically, the corresponding equations (5.3.1), (5.3.3)

and (5.3.5) respectively. For simplicity, in this sub-section we omit the super-script ν and write the approximate solutions as U2, V2, P2.

Proof of Theorem 3.1. We divide the proof into three parts.

Part 1. We first investigate the convergence of U2. Straightforward computa-tions yield

(U2)t =dtνγχ′ν

(U2 − U2

)+ χν · (U2)t + (1− χν)(U2)t, (5.4.7)

(U2)x =1

νγχ′ν

(U2 − U2

)+ χν · (U2)x + (1− χν)(U2)x, (5.4.8)

(U2)xx =1

ν2γχ′′ν

(U2 − U2

)+

2

νγχ′ν

(U2 − U2

)x

+χν · (U2)xx + (1− χν)(U2)xx. (5.4.9)

Hereafter, to write the derivatives of U2 in the form, like the first term in theright hand side of (5.4.7), we changed the arguments t, x of U2, V2, P2 to t, d,where d = d(x, t) is defined by

d(x, t) = x− φ(t).

However, risking abuse of notations we still denote U2(d, t), V2(d, t), P2(d, t) byU2, V2, P2 for the sake of simplicity. After such a transformation of arguments,the terms are easier to deal with, as we shall see later on.

Therefore, we find that U2 satisfies

(U2)t − ν (U2)xx + (F (U2))x = I1 + I2 + I3, (5.4.10)

where Ik (k = 1, 2, 3) are the collections of like-terms according to whether ornot their supports are contained in a same sub-domain of R, more precisely,

they are defined by

I1 = χν

((U2)t − ν(U2)xx + f(U2) · (U2)x

), (5.4.11)

I2 = (1− χν)((U2)t − ν(U2)xx + f(U2)(U2)x

), (5.4.12)

I3 = (U2 − U2)

(dtχ

′ν

νγ− χ′′

ν

ν2γ−1+χ′νf(U2)

νγ

)− 2χ′

ν

νγ−1

(U2 − U2

)x.(5.4.13)

It is easy to see that the support of I1, I2 is, respectively, a subset of[0, 2νγ] and a subset of [νγ,∞), while the support of I3 is a subset of [νγ, 2νγ]∪[−2νγ,−νγ].

Now we turn to estimate I1, I2, I3. Firstly we handle I3. In this case one canapply the matching conditions (5.3.72) – (5.3.74) and use Taylor expansionsto obtain

∂lx(U2 − U2)(x, t) = O(1)ν(3−l)γ (5.4.14)

on the domain (x, t) | νγ ≤ |x − φ(t)| ≤ 2 νγ, 0 ≤ t ≤ T and l = 0, 1, 2, 3.From these estimates (5.4.14), which can also be found, e.g. in Goodman andXin [20], the following assertion follows easily that

I3 = O(1)ν2γ as ν → 0. (5.4.15)

Moreover, we have∫R|I3(x, t)|2dx =

∫νγ≤|x−φ(t)|≤2 νγ

|I3(x, t)|2dx ≤ Cν5γ. (5.4.16)

To deal with I1, I2 we rearrange the terms of I1, I2 as follows

I1 = χν

((U2)t − ν(U2)xx + (F (U2))x

)+ χν ·

(f(U2)− f(U2)

)(U2)x,

= I1a + I1b, (5.4.17)

I2 = (1− χν)((U2)t − ν(U2)xx + (F (U2))x

)+ (1− χν)

(f(U2)− f(U2)

)(U2)x,

= I2a + I2b. (5.4.18)

Moreover, I1b can be rewritten as

I1b = χν

∫ 1

0

f ′(sU2 + (1− s)U2)ds · (U2 − U2)(U2)x. (5.4.19)

Note that supp I1b ⊂ (x, t) ∈ QT | |x − φ(t)| ≤ 2 νγ and U2(x, t) = U2(x, t)if (x, t) ∈ QT | |x − φ(t)| ≤ νγ. Therefore, from (5.4.14) and (5.4.19) weobtain

|I1b| =C

ν|(U2 − U2)(U2)

′| = O(ν3γ−1), (5.4.20)

where we choose γ > 13so that 3γ − 1 > 0. Recalling the construction of U2,

from equations (5.3.69) – (5.3.71), we can rewrite I1a as

I1a = χν

((U2)t − ν(U2)xx + (F (u0) + νf(u0)u1 + ν2(f(u0)u2 +

1

2f ′(u0)u

21) +Ru)x

)= χν(Ru)x, (5.4.21)

where the remainder Ru is defined by

Ru = F (U2)−(F (u0) + νf(u0)u1 + ν2(f(u0)u2 +

1

2f ′(u0)u

21)

)= O(ν3).

Thus

|I1a| ≤ |(Ru)x| =1

ν|(Ru)

′| = O(ν2). (5.4.22)

In a similar manner, we now handle I2, and rewrite I2b as

I2b = (1− χν)

∫ 1

0

f ′(sU2 + (1− s)U2)ds · (U2 − U2)(U2)x. (5.4.23)

It is easy to see that supp I2b ⊂ |d| ≥ νγ and U2 = U2 if |d| ≥ 2νγ. From thefact that U2 − U2 = χν(U2 − U2) and (5.4.14) it follows that

|I2b| ≤ C χν |(U2 − U2)(U2)x| = O(ν3γ). (5.4.24)

As for I2a, invoking equations (5.3.29) and (5.3.30) we assert that there holds

I2a = (1− χν)((U2)t − ν(U2)xx + (F (u0) + νf(u0)u1 +Ru)x

)= (1− χν)(Ru)x +O(ν2). (5.4.25)

and the remainder Ru is given by

Ru = F (U2)− (F (u0) + νf(u0)u1) = O(ν2),

hence

I2a = O(ν2). (5.4.26)

On the other hand, we have∫R|I1(x, t)|2dx =

∫|x−φ(t)|≤2 νγ

|I1(x, t)|2dx

≤∫|x−φ(t)|≤νγ

|I1a(x, t)|2dx+∫νγ≤|x−φ(t)|≤2 νγ

|I1b(x, t)|2dx

≤ C(ν2∗2+1 + ν6γ−2+γ) ≤ C νγ. (5.4.27)

Here we used the simple inequalities: 6γ − 2 + γ < 5 and 6γ − 2 > 0 since weassume γ ∈ (1

3, 1).

Similarly, one can obtain∫R|I2(x, t)|2dx =

∫|x−φ(t)|≥νγ

|I2(x, t)|2dx ≤ C(ν2∗2+1 + ν6γ+γ) ≤ C νγ.(5.4.28)

In conclusion, from (5.4.10), (5.4.15), (5.4.20), (5.4.22), (5.4.24) and (5.4.26)we are in a position to assert that U ν

2 satisfies the equation in the followingsense

(U ν2 )t − ν(U ν

2 )xx + (F (U ν2 ))x = O(να),

as ν → 0. Here α = 3γ − 1 and we used the fact that 3γ − 1 < 2γ < 2by assumption γ < 1. Furthermore, from construction we see easily that theinitial data is satisfied asymptotically too.

Part 2. We now turn to investigate the convergence of V2. Similar computa-tions show that V2 can be written in terms of V2, V2 as

(V2)t =dtνγχ′ν

(V2 − V2

)+ χν · (V2)t + (1− χν)(V2)t, (5.4.29)

(V2)x =1

νγχ′ν

(V2 − V2

)+ χν · (V2)x + (1− χν)(V2)x, (5.4.30)

(V2)xx =1

ν2γχ′′ν

(V2 − V2

)+

2

νγχ′ν

(V2 − V2

)x

+χν · (V2)xx + (1− χν)(V2)xx, (5.4.31)

and V2 satisfies the following equation

(V2)t − ν (V2)xx + (f(U2)V2)x = J1 + J2 + J3, (5.4.32)

where Jk (k = 1, 2, 3) are given, according to their supports, by

J1 = χν

((V2)t − ν(V2)xx +

(f(U2)V2

)x

), (5.4.33)

J2 = (1− χν)((V2)t − ν(V2)xx +

(f(U2)V2

)x

), (5.4.34)

J3 =

(dtχ

′ν

νγ− χ′′

ν

ν2γ−1+f(U2)χ

′ν

νγ

)(V2 − V2)−

2χ′ν

νγ−1(V2 − V2)x.(5.4.35)

Since for V2, we also have the same estimate (5.4.14) which is valid for U2,namely we have

∂lx(V2 − V2) = O(1)ν(3−l)γ (5.4.36)

on the domain (x, t) | νγ ≤ |x − φ(t)| ≤ 2 νγ, 0 ≤ t ≤ T and l = 0, 1, 2, 3.Thus it follows from (5.4.36) and the uniform boundedness of U2 that

J3 = O(1)ν2γ, as ν → 0. (5.4.37)

The investigation of convergence for J1, J2 is more technically complicatedthan I1, I2. Rewrite J1, J2 as

J1 = χν

((V2)t − ν(V2)xx + (f(U2)V2)x

)+ χν ·

((f(U2)− f(U2)) V2

)x

= J1a + J1b, (5.4.38)

and

J2 = (1− χν)((V2)t − ν(V2)xx + (f(U2)V2)x

)+ (1− χν)

((f(U2)− f(U2)V2

)x

= J2a + J2b. (5.4.39)

We now deal with J1b which can be changed to

J1b = χν

(∫ 1

0

f ′(sU2 + (1− s)U2)ds(U2 − U2) V2

)x

= χν

(∫ 1

0

f ′(sU2) + (1− s)U2)ds V2

)x

(U2 − U2)

+χν

∫ 1

0

f ′(sU2) + (1− s)U2)ds V2

(U2 − U2

)x

= O(ν3γ−1) +O(ν2γ) = O(ν3γ−1), (5.4.40)

here we used that 3γ − 1 < 2γ since γ < 1.Rewriting J1a as

J1a = χν

((V2)t − ν(V2)xx +

((f(u0) + f ′(u0)(νu1 + ν2u2) +

1

2f ′′(u0)(νu1)

2)V2

)x

)+χν(RvV2)x

= χν

(O(ν2) + (RvV2)x

). (5.4.41)

Here, equations (5.3.76) - (5.3.78) were used. And the remainder Rv is

Rv = f(U2)− (f(u0) + f ′(u0)(νu1 + ν2u2) +1

2f ′′(u0)(νu1)

2) = O(ν3)

Therefore, from (5.4.41) one has

J1a = χν

(O(ν2) + (RvV2)x

)= O(ν2). (5.4.42)

The terms J2a, J2b can be estimated in a similar way and we obtain

J2b = O(ν3γ) +O(ν2γ) = O(ν2γ), (5.4.43)

and

J2a = (1− χν)((V2)t − ν(V2)xx + ((f(u0) + νf ′(u0)u1 +Rv)V2)x

)= (1− χν)

(O(ν2) + (RvV2)x

),

where Rv is given by Rv = f(U2)− (f(u0) + νf ′(u0)u1) = O(ν2). It is easy tosee that

J2a = O(ν2). (5.4.44)

On the other hand, we have the following estimates of integral type∫R|J1(x, t)|2dx =

∫|x−φ(t)|≤2 νγ

|J1(x, t)|2dx

≤∫|x−φ(t)|≤νγ

|J1a(x, t)|2dx+∫νγ≤|x−φ(t)|≤2 νγ

|J1b(x, t)|2dx

≤ C(ν2∗2+1 + ν6γ−2+γ) ≤ C νγ. (5.4.45)

and∫R|J2(x, t)|2dx =

∫|x−φ(t)|≥νγ

|J2(x, t)|2dx ≤ C(ν2∗2+1 + ν4γ+γ) ≤ C νγ.(5.4.46)

Therefore, it follows from (5.4.32), (5.4.37), (5.4.40), (5.4.42), (5.4.43) and(5.4.44) that V ν

2 satisfies the equation in the following sense

(V ν2 )t − ν(V ν

2 )xx + (f(U ν2 )V

ν2 )x = O(ν3γ−1),

as ν → 0. By construction, the initial data is satisfied asymptotically as well.

Part 3. Finally we turn to investigate the convergence of P2. Computationsshow that the derivatives of P2 can be written in terms of P2, P2 as

(P2)t =dtνγχ′ν

(P2 − P2

)+ χν · (P2)t + (1− χν)(P2)t, (5.4.47)

(P2)x =1

νγχ′ν

(P2 − P2

)+ χν · (P2)x + (1− χν)(P2)x, (5.4.48)

(P2)xx =1

ν2γχ′′ν

(P2 − P2

)+

2

νγχ′ν

(P2 − P2

)x

+χν · (P2)xx + (1− χν)(P2)xx, (5.4.49)

and P2 satisfies the following equation

−(P2)t − ν (P2)xx − f(U2)(P2)x = K1 +K2 +K3, (5.4.50)

where Ki (i = 1, 2, 3) are given, according to their supports, by

K1 = −χν

((P2)t + ν(P2)xx + f(U2)(P2)x

), (5.4.51)

K2 = −(1− χν)((P2)t + ν(P2)xx + f(U2) (P2)x

), (5.4.52)

K3 = −

(dtχ

′ν

νγ+

χ′′ν

ν2γ−1+f(U2)χ

′ν

νγ

)(P2 − P2)−

2χ′ν

νγ−1(P2 − P2)x.(5.4.53)

By similar arguments as done for U ν2 , we can prove that

−(P ν2 )t − ν(P ν

2 )xx − f(U ν2 )(P

ν2 )x = O(ν3γ−1),

as ν → 0.

5.4.2 Proof of the convergence

This sub-section is devoted to the proof of Theorem 4.1. Since this sub-sectionis concerned with the proof of convergence as ν → 0, we denote U ν

2 , Vν2 , P

ν2

by U ν , V ν , P ν , respectively, for the sake of simplicity. We begin with thefollowing lemma.

Lemma 5.4.2 For η defined in (5.4.5),

sup0≤t≤T

∥uν(·, t)− u(·, t)∥+ sup0≤t≤T

∥vν(·, t)− v(·, t)∥ ≤ Cνη, (5.4.54)

sup0≤t≤T

∥pν,n(·, t)− p(·, t)∥L1loc(R) → 0, (5.4.55)

as ν → 0, then n → ∞. Here pν,n denotes the solution to smoothed adjointproblem (5.3.5) – (5.3.6).

Proof. Firstly, by construction of the approximate solutions, we conclude thati) for (x, t) ∈ |x− φ(t)| ≥ 2νγ, 0 ≤ t ≤ T,

Uν2 (x, t) = u(x, t) +O(1)ν,

where O(1) denotes a function which is square-integrable over outer region bythe argument in sub-section 4.1.ii) for (x, t) ∈ |x− φ(t)| ≤ νγ,

U ν2 (x, t) = u0(x, t) +O(1)νγ,

iii) and for (x, t) ∈ νγ ≤ |x− φ(t)| ≤ 2νγ,

U ν2 (x, t) = U ν

2 (x, t) + χν(x, t)(Uν2 (x, t)− U ν

2 (x, t)) ∼ U ν2 (x, t) +O(1)ν3γ.

Here we have used again the estimate U ν2 (x, t) − Uν

2 (x, t) = O(1)ν3γ. We canalso establish similar estimates for V ν

2 , Pν2 . Therefore, one can obtain

sup0≤t≤T

∥u(·, t)− U ν(·, t)∥2 + sup0≤t≤T

∥v(·, t)− V ν(·, t)∥2 ≤ Cνmin3γ,2,(5.4.56)

sup0≤t≤T

∥pn(·, t)− P ν(·, t)∥2 ≤ Cνmin3γ,2,(5.4.57)

where pn is the reversible solution to the inviscid adjoint equation −∂tp −f(u)∂xp = 0 with final data p(x, T ) = pTn (x).

By Theorem 4.1.10 which is concerned with the stability (with respect tocoefficient and final data) of reversible solutions in [6] (see also Theorem 5.1in the appendix of this chapter), and the assumptions of final data pn and theone-sided Lipschitz condition we conclude that the reversible solution pn of thebackward problem with initial data pn satisfies

pn → p in C([0, T ]× [−R,R])

for any R > 0, here p is the reversible solution to −pt − f(u)∂xp = 0, inR× (0, T ) with finial data p(x, T ) = pT (x) for x ∈ R. Therefore, we have

sup0≤t≤T

∥pn − p∥L1loc(R) → 0, (5.4.58)

as n→ ∞.Secondly, we need to estimate sup0≤t≤T ∥uν(·, t) − U ν(·, t)∥2, etc. Suppose

that we have obtained

sup0≤t≤T

∥uν(·, t)− U ν(·, t)∥2 ≤ Cνη1 (5.4.59)

where η1 = γ + 1. Then we arrive at (5.4.54) by using the triangle inequality,the estimate (5.4.56) and the fact that min

32γ, 1+γ

2, 1= min

32γ, 1+γ

2

(since

13< γ < 1)). So we conclude for v, p.

Part 1. Now we prove (5.4.59). To this end, we define

w(x, t) = uν(x, t)− U ν(x, t),

Then w(x, t) satisfies

wt − νwxx + f(uν)wx = F , (5.4.60)

w(x, 0) = w0(x). (5.4.61)

Here

F := −3∑

i=1

Ii −Q(x, t)− f ′(U ν)wU νx (5.4.62)

Q :=(f(uν)− f(U ν)− f ′(U ν)w

)U νx . (5.4.63)

Rescaling as follows

w(x, t) = νw(y, τ), where y =x− φ(t)

ν, and τ =

t

ν,

one has

wt(x, t) = wτ (y, τ)−φ′(ντ)wy, wx(x, t) = wy(y, τ), wxx(x, t) =1

νwyy(y, τ),

and problem (5.4.60) – (5.4.61) turns out to be

wτ − wyy + (f(uν)− φ′(ντ))wy = F(νy + φ, ντ), (5.4.64)

w(x, 0) = w0(x). (5.4.65)

Here the initial data w0 can be chosen very small such that

∥w0∥2H1(R) ≤ Cν ≤ Cνγ,

provided that ν is suitably small.The existence of solution w ∈ C([0, T/ν];H1(R)) to problem (5.4.64) –

(5.4.65) follows from the method of continuation of local solution, which is ba-sed on local existence of solution and a priori estimates stated in the followingproposition

Proposition 5.4.3 (A priori estimates) Suppose that problem (5.4.64) – (5.4.65)has a solution w ∈ C([0, τ0];H

1(R)) for some τ0 ∈ (0, T/ν]. There exists posi-tive constants µ1, ν1 and C, which are independent of ν, τ0, such that if

ν ∈ (0, ν1], sup0≤τ≤τ0

∥w(τ, ·)∥H1(R) + µ0 ≤ µ1, (5.4.66)

then

sup0≤τ≤τ0

∥w(τ, ·)∥2H1(R) +

∫ τ0

0

∥w(τ, ·)∥2H2(R)dτ ≤ Cνγ.

Proof. Step 1. By the maximum principle and construction of approximatesolution U ν , we have

∥uν∥L∞(Qτ0 )≤ C, ∥U ν∥L∞(Qτ0 )

≤ C. (5.4.67)

Whence, from the smallness assumption (5.4.66) and definition of Q it followsthat

|Q(x, t)| ≤ C|w2U νx |. (5.4.68)

Step 2. Multiplying eq. (5.4.64) by w and integrating the resulting equa-tion with respect to y over R we obtain

1

2

d

dτ∥w∥2 + ∥wy∥2 +

∫R(f(uν)− φ′(ντ))wywdy =

∫RF(νy + φ, ντ)wdy.(5.4.69)

We first deal with the term∫Rf ′(U ν)wU ν

xwdy =

∫Rf ′(U ν)w2νU ν

xdy.

From the property of profile U we have

νU νx = Uν

y → 0, as ν → 0.

Thus ∣∣∣∣∫Rf ′(U ν)wU ν

xwdy

∣∣∣∣ ≤ C∥w∥2.

For the term of Q, it is easier to obtain that∣∣∣∣∫RQ(y, τ)U ν

xwdy

∣∣∣∣ ≤ C

∫R|w2U ν

xw|dy ≤ C∥w∥2.

It remains to deal with the term∫R Iwdy where I =

∑3i=1 Ii. We invoke the

L2-norm estimates for I, i.e. (5.4.27) and (5.4.28), which have been obtainedin Subsection 3.1, and get∣∣∣∣∫ τ

0

∫RIwdτdy

∣∣∣∣ ≤ C

(∫ τ

0

∫R|I|2dτdy +

∫ τ

0

∫R|w|2dτdy

)≤ Cνγ + C

∫ τ

0

∫R|w|2dτdy. (5.4.70)

Finally, by the Young inequality one gets∣∣∣∣∫R(f(uν)− φ′(ντ))wywdy

∣∣∣∣ ≤ 1

2∥wy∥2 + C∥w∥2.

Therefore, the above arguments and integrating (5.4.69) with respect to τyield

1

2∥w(τ)∥2 + 1

2

∫ τ

0

∥wy(s)∥2ds ≤ C

∫ τ

0

∥w(s)∥2ds+ Cνγ, (5.4.71)

from which and the Gronwall inequality in the integral form we then arrive at

∥w(τ)∥2 ≤ Cνγ.

So we also obtain

∥uν(·, t)− U ν(·, t)∥2 = ν∥w(τ)∥2 ≤ Cν1+γ.

Step 3. Next we multiply eq. (5.4.65) by −wyy and integrate the resultantwith respect to y to get

1

2

d

dτ∥wy∥2 + ∥wyy∥2 −

∫R(f(uν)− φ′(ντ))wyywydy

= −∫RF(νy + φ, ντ)wyydy, (5.4.72)

Using (5.4.67) and estimates on Ii (I = 1, 2, 3), from the Young inequality wethen arrive at

1

2∥wy(τ)∥2 +

1

2

∫ τ

0

∥wyy(s)∥2ds ≤ C

∫ τ

0

∥wy(s)∥2ds+ Cνγ, (5.4.73)

making use of Gronwall’s inequality again one has

∥wy(τ)∥2 ≤ Cνγ. (5.4.74)

Part 2. In this part we are in a position to prove the convergence of pν,n− P ν ,namely to prove

sup0≤t≤T

∥pν,n(·, t)− P ν(·, t)∥2 ≤ Cνη. (5.4.75)

Letq = pν,n − P ν ,

then computations yield that q satisfies

−qt − νqxx − f(uν)qx = G, (5.4.76)

q(x, 0) = q0(x). (5.4.77)

Here

G := −3∑

i=1

Ki −Q1(x, t)− f ′(U ν)wP νx (5.4.78)

Q1 :=(f(uν)− f(U ν)− f ′(U ν)w

)P νx , (5.4.79)

and Ki (with i = 1, 2, 3) can be defined in a slightly different way as in thearguments for V ν .

Rescaling again as follows

q(x, t) = νq(y, τ), where y =x− φ(t)

ν, and τ =

T − t

ν,

one has

qt(x, t) = −qτ (y, τ)− φ′(ντ)qy, qx(x, t) = qy(y, τ), qxx(x, t) =1

νqyy(y, τ),

and problem (5.4.76) – (5.4.77) can be rewritten as

qτ − qyy − (f(uν)− φ′(ντ)) qy = G(νy + φ, ντ), (5.4.80)

q(x, 0) = q0(x). (5.4.81)

Employing again the method of continuation of a local solution based upon apriori estimates, we prove easily the existence of solution q ∈ C([0, τ0];H

2(R)),also we have

∥pν,n(·, t)− P ν(·, t)∥2 = ν∥q(τ)∥2 ≤ Cν1+γ,

here we rewrite pν as pν,n to indicate that solution depends on n too. Thisestimate implies that

sup0≤t≤T

∥pν,n(·, t)− pn(·, t)∥L1loc(R) → 0,

invoking (5.4.58) we obtain

sup0≤t≤T

∥pν,n(·, t)− p(·, t)∥L1loc(R) → 0,

when ν → 0 firstly, then n→ ∞.Furthermore, we assume that the initial p|t=0 is bounded in H2(R), similar

to the argument in Goodman and Xin [20], we can prove that for any constanth > 0

sup0≤t≤T

∥pν,n − pn∥L∞(Ωh) → 0, (5.4.82)

as ν → 0, here Ωh is defined by

Ωh = (x, t) ∈ QT | |x− φ(t)| > h.

On the other hand, from the stability (of the reversible solution) theorem 4.1.10in [6], we have

sup0≤t≤T

∥pn − p∥L∞(Ωh) → 0, (5.4.83)

as n→ ∞. Thus there holds

sup0≤t≤T

∥pν,n − p∥L∞(Ωh) → 0, (5.4.84)

as ν → 0, then n→ ∞.

Part 3. To prove the convergence of O := vν− V ν2 , we rewrite equations (5.3.3)

and (5.3.104) as follows

vνt − νvνxx + f(uν)(vν)x + (f(uν))xvν = 0, (5.4.85)

(V ν2 )t − ν(V ν

2 )xx + f(Uν2 )(V

ν2 )x + (f(U ν

2 ))xVν2 =

3∑i=1

Ji. (5.4.86)

Then we find O satisfies

Ot − νOxx + f(uν)Ox + (f(uν))xO = H, (5.4.87)

and

H = −(f(uν)− f(U ν

2 ))xV ν2 −

(f(uν)− f(U ν

2 ))(V ν

2 )x −3∑

i=1

Ji.

We use again the rescaling technique as follows

O(x, t) = νO(y, τ), where y =x− φ(t)

ν, and τ =

t

ν,

To overcome the difficulty, due to the last term in the left hand side of(5.4.87), in the proof of the convergence of O, we make use of the convergenceresult in Part 1, the estimate ν

∫ t

0∥uνx(τ)∥2dτ ≤ C, the interpolation inequality

in the following form

∥f∥L4(R) ≤ C∥fx∥14∥f∥

34 + C ′∥f∥,

and the Young inequality of the form: abc ≤ εa4 + Cε(b4 + c2). We then

estimate ∣∣∣∣∫R(f(uν))xOOdx

∣∣∣∣ ≤ Cν

∫R|uνx|O2dx

≤ Cν∥uνx∥∥O∥2L4

≤ Cν∥uνx∥(∥Ox∥24∥O∥

32 + C ′∥O∥2)

≤ C(ν2∥uνx∥2∥O∥2 + ∥O∥2) + 1

2∥Ox∥2. (5.4.88)

The last term in the right hand side of (5.4.88) can be absorbed by the lefthand side. The other terms in (5.4.87) and H can be treated in a similar wayas in Part 1, and we omit the details.

Therefore, the proof of Theorem 4.1 is complete.

5.5 The method of alternating descent direc-

tions: Viscous case

This section is concerned with the extension of the arguments in sub-section 2.3on the choices of alternating descent directions, to the viscous problem. Weconsider the case that the initial data uI satisfies

uI is continuous and smooth up to the shock, and uIx has only adiscontinuity at φI . Moreover, uI , uIx are integrable over QT\Σ.

(5.5.1)

As pointed out in the end of Section 1, for any positive ν solutions δu, p ofequations (5.1.6) and (5.1.10) are smooth, thus the Gateaux derivative of thefunctional J is as follows

δJ = < δJ(uI), δuI > =

∫Rp(x, 0)δuI(x)dx, (5.5.2)

where the adjoint state p = pν is the solution to (5.1.10) with initial datump(x, T ) = u(x, T )− uD(x).

To exploit the possibilities that the alternate descent method provides, wetake into account, as in [12], the effects of possible infinitesimal perturbationsof initial datum and also infinitesimal translations, and choose the initial datauIε of the form

uIε(x) = uIε(x+ εδφI) + εδuIε(x), (5.5.3)

By a Taylor expansion, (5.5.3) can be rewritten in the following form

uIε(x) = uI(x) + ε(δφIuIx(x) + δuI(x)

)+O(ε2). (5.5.4)

Correspondingly we formulate the linearized problem as follows

(δu)t + (f(u)δu)x = ν(δu)xx, (5.5.5)

δu(x, 0) = δφIuIx(x) + δuI(x), (5.5.6)

and its adjoint problem is

−pt − f(u)px = νpxx, (5.5.7)

p(x, 0) = pTn (x). (5.5.8)

But by doing this way it leads to some difficulties: A Dirac delta appears inthe Taylor expansion (5.5.4) and the initial data (5.5.6) in the case uI(x) hasa jump. How to understand this expansion and how to solve problem (5.5.5)– (5.5.6)? This initial value problem is difficult even (5.5.5) is parabolic for

fixed ν. There are not too many references related to this topic: some authorsinvestigate generalized solutions to Burgers’ equation with singular data see,e.g. [4], while in [10, 5, 13] the authors studied parabolic equations with a Diracdelta as an initial datum. However the solution exists only for some specialcases. Another difficulty is that we need more regular initial data for ourconstruction of asymptotic expansions, moreover, we also consider the limit asν → 0, the limit equation of (5.5.5) has discontinuous coefficient, which leadsto a term like (f(u)δu)x where f(u) and δu may be discontinuous.

Therefore, we don’t expand uI(x + εδφI) directly as done in (5.5.4). Toovercome the above difficulties we approximate the initial datum as follows:

We use again the cut-off function χh for h > 0, define ξ = x−φI

hand choose U I

a smooth function in ξ satisfying the matching conditions

limξ→±∞

U I(ξ) = limξ→φI±0

uI(x),

then by a Taylor expansion, we obtain

uIε,h(x) = χh(x)UI(ξ) + (1− χh(x))u

I(x+ εδφI) + εδuIε(x)

= χh(x)UI(ξ) + (1− χh(x))u

I(x)

+ε((1− χh(x))u

Ix(x)δφ

I + δuIε(x))+O(ε2), (5.5.9)

Letting h→ 0 we see that there is no any Dirac delta appearing in (5.5.9) anymore. The corresponding linearized problem turn out to

(δu)t + (f(u)δu)x = ν(δu)xx, (5.5.10)

δu(x, 0) = δφI(1− χh(x))uIx(x) + δuI(x), (5.5.11)

Moreover for any fixed ν we can easily pass the solution δuν,h of problem(5.5.10) – (5.5.11) to its limit δuν as h→ 0.

We assume, to begin with, that uI satisfies (5.5.1). We shall make use of theconvergence result (5.4.84), from which one concludes that the smooth solutionpν,n of problem (5.5.7)–(5.5.8) is very close to its limit p, provided that ν isvery small and n is very large. To determine the alternating descent directions,the first thing to be done is to identify the region of influence [x−, x+] of theinner boundary of the inviscid adjoint system. Similar to [12], we can computex−, x+, so the region of influence is thus defined. Then we need to identify thevariations (δuI , δφI) such that∫ x+

x−pν,n(x, 0)

(δφI(1− χh(x))u

Ix(x) + δuI(x)

)dx = 0. (5.5.12)

It is easy to see that we can rewrite (5.5.13) as∫[x−,x+]\φI

pν,n(x, 0)(δφIuIx(x) + δuI(x)

)dx = oh(1), (5.5.13)

where, oh(1) denotes a small quantity such that oh(1) → 0 as h→ 0.In [12], it is argued that if pν,n(x, 0) were constant within the interval [x−, x+]

as in the inviscid case, this would amount to consider variations such that

δφI = −∫ x+

x− δuI(x)dx

uI(x+)− uI(x−). (5.5.14)

One possibility would be to consider variations δuI in [x−, x+] such that∫ x+

x− δuI(x)dx = 0 and δφI = 0. The variation of the functional J wouldthen be

δJ =

∫x>x+∪x<x−

pν,n(x, 0)δuI(x)dx,

and the optimal descent direction

δuI(x) = −pν,n(x, 0), in x > x+ ∪ x < x−.However, the assumption that pν,n(x, 0) is constant within the interval

[x−, x+], is not true, in general, in the viscous case, and is only true in theinviscid case. Invoking (5.4.84), we find pν,n(x, 0) in (5.5.13) is close to aconstant over Ωµ, provided that ν is small and n is large. Thus we rewrite(5.5.13) as follows

oh(1) =

∫ x+

x−pν,n(x, 0)

(δφIuIx(x) + δuI(x)

)dx

=

∫ φI+µ

φI−µ

+

∫[x−,x+]\[φI−µ,φI+µ]

pν,n(x, 0)(δφIuIx(x) + δuI(x)

)dx

= I1 + I2. (5.5.15)

Here, µ is a small positive number. Recalling δuI ∈ L1(R) ∩ L∞(R), by as-sumptions (5.5.1) and the results pν,n ∈ L∞(QT ), we see that the integrand isintegrable over R\φI, then I1 is small and depends on the small parameterµ (also n, ν, but they are assumed temporarily to be fixed). While for I2 wecan replace pν,n by p, however a small error, depending on ν and n, appears.Then (5.5.15) can be rewritten as

0 = Ch,ν,µ,n +

∫[x−,x+]\[φI−µ,φI+µ]

p(x, 0)(δφIuIx(x) + δuI(x)

)dx

= Ch,ν,µ,n + p(x, 0)(uI(x+)− uI(x−)−

(uI(φI + µ)− uI(φI − µ)

))δφI

+p(x, 0)

(∫[x−,x+]\[φI−µ,φI+µ]

δuI(x)dx

). (5.5.16)

Here Ch,ν,µ,n denotes a small quantity depending on h, ν, µ, n. But by assump-tion (5.5.1), uI(φI + µ)− uI(φI − µ) → [uI ]φI as µ→ 0. Therefore,

δφI = −Ch,ν,µ,n/p(x, 0) +

∫[x−,x+]\[φI−µ,φI+µ]

δuI(x)dx

uI(x+)− uI(x−)− [uI ]φI

∼ −

∫[x−,x+]

δuI(x)dx

uI(x+)− uI(x−)− [uI ]φI

. (5.5.17)

This implies that we can choose a descent direction as the case that pν,n(x, 0) isa constant, at least for numerical simulation since there are some errors whenwe compute any quantity. Moreover, we can extend δuI to the sub-domain[x−, x+] such that ∫ x+

x−δuI(x)dx = 0,

whenceδφI = 0.

The second class of variations is the one that takes advantage of the in-finitesimal translations δφI . We can then set δuI ≡ 0 and choose δφI suchthat

δφI = −∫R\φI

p(x, 0)uIx(x)dx− [uI ]φIp(φI , 0).

As mentioned above, we could consider slightly different variations of the initialdata of the form

δφI = −[uI ]φIp(φI , 0)

as in [11].In this way, we have identified two classes of variations and its approximate

values inspired in the structure of the state and the adjoint state in the inviscidcase, allowing to implement the method of alternating descent in the inviscidcase when uI is discontinuous.

The efficiency of the method discussed here has been illustrated by severalnumerical experiments in the case of the Burgers equation in reference [12],where an implicit assumption that σ is large is assumed due to the use ofequation (5.2.6) (corresponding to (5.3.63) in the case that σ = ∞) withf(u) = u. However, with the help of the modified equation (5.3.63), we cancarry out simulations for optimal control problems of nonlinear conservationlaws, in the case that σ is not too large.

From the above arguments we can draw the following conclusion:Conclusion: There exists a number ν0 such that for any ν ∈ (0, ν0], p(x, 0)

can be used to replace the exact solution pν,n(x, 0) (as in (5.5.16)) with a smallerror which is much smaller than the mesh size. Thus this error can be omitted,and the algorithm (see Algorithm 6 in [11]) of the alternating descent methodfor inviscid Burgers equation is applicable to the viscous problem with smallviscosity, and this method is efficient.

If ν ∈ (ν0,∞), then the solutions uν , δuν are smooth for any t > 0 and pν,nis smooth for t < T . In this case, if we replace the exact solution pν,n(x, 0) byp(x, 0), the error is probably not sufficiently small, so the alternating descentmethod is not efficient in this case, instead the classical descent method isapplicable.

Appendix

For the convenience of the reader, we record the definition of the reversiblesolutions to linear transport equation and Theorem 4.1.10 in [6].

Let SLip = Liploc([0, T ]× R). Denote by L the solutions p ∈ SLip to

∂tp+ a∂xp = 0.

We study solutions to the backward problem, consisting of all p ∈ L such thatp(·, T ) = pT for any given pT ∈ Liploc(R).

Definitions (Nonconservative reversible solutions) i) We call exceptional so-lution any function pe ∈ L such that pe(·, T ) = 0. We denote by E the vectorspace of exceptional solutions.

ii) We call domain of support of exceptional solutions the open set

Ve = (x, t) ∈ R× (0, T ) | ∃pe ∈ E , pe(x, t) = 0.

iii) Any p ∈ L is called reversible if p is locally constant in Ve.

The following is an important feature of reversible solutions, namely thestability with respect to coefficient and final data:

Theorem 5.5.1 (Stability) Let (an) be a bounded sequence in L∞(R×(0, T ))such that an

∗ a in L∞(R × (0, T )). Assume that ∂xan ≤ αn(t), where (αn)is bounded in L1((0, T )), ∂xa ≤ α where α ∈ L1((0, T )). Let (pTn ) be a boundedsequence in Liploc(R), pTn → pT , and denote by pn the reversible solution to

∂tpn + an∂xpn = 0, in R× (0, T ),

pn(x, T ) = pTn (x).

Then pn → p in C([0, T ] × [−R,R]) for any R > 0, where p is the reversiblesolution to

∂tp+ a∂xp = 0, in R× (0, T ),

p(x, T ) = pT (x).

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Index

asymptotic expansion, i, 3

a priori estimate, 69adjoint problem, 33adjoint state pair, 40alternating descent direction, 74alternating descent method, i, 31ansatz, 6approximate solution, 2, 58asymptotic analysis, i, 2asymptotic approximation, 3asymptotic sequence, 3averaging method, 3

boundary layer, 17, 24, 27boundary layers, 16Burgers equation, 31

classification of the generalized tangentvectors, 41

common part, 18conclusion, 77conservation law, i, 29convention, 47convergence, 60cost functional, 32cut-off function, 19, 75

decomposition, 28derivation of the interface equations,

52descent direction, 43

entropy solution, 32exceptional solution, 78

existence of minimizers, 45expansion

asymptotic expansionmatched, i, 25

multi-scale, 3

fast variable, 22Fife, 18, 25

gauge function, 3generalized Gateaux derivative, 39generalized tangent vector, 37

infinitesimal perturbation, 45infinitesimal translation, 45inner expansion, 17, 21, 55intermediate region, 18iterative method, 7

linearized problem, 33, 74

matching byintermediate variable, 25Van Dyke’s rule, 26

matching condition, 16, 18, 24, 25, 57

non-ingeral powers, 11

Oleinik’s one-sided Lipschitz condition,47

optimal control, i, 31orthogonality condition, 56outer expansion, 16, 21, 47

perturbationregular, 5

84

singular, 2, 8perturbation method, 3Poincare, 2Prandtl, 2problem

regular, 13singular, 13

profile, 11

Rankine-Hugoniot condition, 36, 53region

inner, 16, 24intermediate, 24matching, 24outer, 16, 24overlapping, 24

region of influence, 75rescaling, 9, 22, 69, 73reversible solution, 34, 41, 72, 78

sensitivity analysis, 36sensitivity in presence of shocks, 39stability of reversible solutions, 78Stieltjes, 2Stirling, 1symbol

Du Bois Reymond, 1Landau, 1

Taylor expansion, 74transport equation, 78

vanishing viscosity method, i, 29variation of shock position, 34

WKBJ approximation, 3