Met a Stability in Burgers Equation

8/8/2019 Met a Stability in Burgers Equation

1/22

arXiv:0810

.0086v1

[math.DS

]1Oct2008

Using global invariant manifolds to understand

metastability in Burgers equation with small viscosity

Margaret Beck

Division of Applied MathematicsBrown University

Providence, RI 02912, USA

C. Eugene Wayne

Department of Mathematics and Center for BioDynamicsBoston University

Boston, MA 02215, USA

October 1, 2008

Abstract

The large-time behavior of solutions to Burgers equation with small viscosity is de-scribed using invariant manifolds. In particular, a geometric explanation is provided fora phenomenon known as metastability, which in the present context means that solutionsspend a very long time near the family of solutions known as diffusive N-waves beforefinally converging to a stable self-similar diffusion wave. More precisely, it is shown thatin terms of similarity, or scaling, variables in an algebraically weighted L2 space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationarysolutions. Through each of these fixed points there exists a one-dimensional, global, at-tractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability

corresponds to a fast transient in which solutions approach this metastable manifold ofdiffusive N-waves, followed by a slow decay along this manifold, and, finally, convergenceto the self-similar diffusion wave.

email: Margaret [email protected]; The majority of this work was done while MB was affiliated with theDepartment of Mathematics, University of Surrey, Guildford, GU2 7XH, UK

email: [email protected]

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1 Introduction

It is well known that viscosity plays an important role in the evolution of solutions to viscousconservation laws and that its presence significantly impacts the asymptotic behavior of solu-tions. Much work has been done to understand the relationship between solutions for zero and

nonzero viscosity. For an overview, see, for example, [Daf05, Liu00]. With regard to Burgersequation, one key property is the following. If u = u(x, t) denotes the solution to Burgersequation with viscosity and u0 = u0(x, t) denotes the solution to the inviscid equation, thenit is known that u u0 in an appropriate sense for any fixed t > 0 as 0. However, forfixed , the large time behavior of u and u0 is quite different, and they converge to solutionsknown as diffusion waves and N-waves, respectively. Thus, the limits 0 and t are notinterchangeable.

Recently, a phenomenon known as metastability has been observed in Burgers equation withsmall viscosity on an unbounded domain [KT01]. Generally speaking, metastable behavior iswhen solutions exhibit long transients in which they remain close to some non-stationary state

(or family of non-stationary states) for a very long time before converging to their asymptoticlimit. In [KT01], the authors observe numerically that solutions spend a very long time near afamily of solutions known as diffusive N-waves, before finally converging to the stable familyof diffusion waves. This terminology1 is due to the fact that the diffusive N-waves are closeto inviscid N-waves. In [KT01] this is proven in a pointwise sense. Furthermore, in terms ofscaling, or similarity, variables, they compute an asymptotic expansion for solutions to Burgersequation with small viscosity. They find that the stable diffusion waves correspond to the firstterm in the expansion, whereas the diffusive N-waves correspond to taking the first two terms.Thus, by characterizing the metastability in terms of these diffusive N-waves, they provide away of understanding the interplay between the limits 0 and t .

In this paper, we show that the metastable behavior in the viscous Burgers equation, describedin [KT01], can be viewed as the approach to, and motion along, a normally attractive, invariantmanifold in the phase space of the equation. In terms of the similarity variables, we show that onehas the following picture. There exists a global, one-dimensional center manifold of stationarysolutions corresponding to the self-similar diffusion waves. Through each of these fixed pointsthere exists a global, one-dimensional, invariant, normally attractive manifold corresponding tothe diffusive N-waves. For almost any initial condition, the corresponding solution of Burgersequation approaches one of the diffusive N-wave manifolds on a relatively fast time scale: =O(| log |). Due to attractivity, the solution remains close to this manifold for all time andmoves along it on a slower time scale, = O(1/), towards the fixed point which has the sametotal mass. Note this this corresponds to an extremely long timescale t O(e1/) in the originalunscaled time variable. This scenario is illustrated in Figure 1.

Note that, in [Liu00], it was shown that the large time behavior of solutions to a general class ofconservation laws is governed by that of solutions to Burgers equation. Roughly speaking, this isdue to the marginality of the nonlinearity in the case of Burgers equation and the fact that any

1These diffusive N-waves are also discussed in [Whi99, 4.5], where they are referred to simply as N-waves.Here, as in [KT01], we reserve the term N-wave for solutions of the inviscid equation and diffusive N-wave forsolutions of the viscous equation.

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)|

Diffusive Nwaves

"Arbitrary" trajectory

Fast transient:

metastable

region

Invariant, normally attractive, manifold

Selfsimilar solutions

centermanifold of

fixed points

(1/)

O(| log(

Figure 1: A schematic diagram of the invariant manifolds in the phase space of Burgers equation,(2.3), and their role in the metastable behavior. The solution trajectory (blue) experiences aninitial fast transient of =

O(

|log

|) before entering a neighborhood of the manifold of diffusive

N-waves (green). It then remains in this neighborhood for all time as it approaches, on the slowertime scale of = O(1/), a point on the manifold of stable stationary states (red).

higher order nonlinear terms in other conservation laws are irrelevant. Therefore, the presentanalysis for Burgers equation could potentially be used to predict and understand metastabilityin other conservation laws with small viscosity, as well.

We remark that a similar metastable phenomenon has also been investigated in Burgers equationon a bounded interval [BKS95, SW99] and numerically observed in the Navier-Stokes equationson a two-dimensional bounded domain [YMC03]. Furthermore, metastability has been observed

in reaction-diffusion equations: for example, on a bounded interval [FH89, CP89, CP90] andspatially discrete lattice [GVV95]. In [CP90], the metastable states were described in terms ofglobal unstable invariant manifolds of equilibria.

The remainder of the paper is organized as follows. In 2 we state the equations and functionspaces within which we will work, as well as some preliminary facts about the existence ofinvariant manifolds in the phase space of Burgers equation. We also precisely formulate ourresults, Theorems 1 and 2. In sections 3, 4, and 5 we prove Lemma 2.1, Theorem 1, andTheorem 2, respectively. Concluding remarks are contained in 6. Finally, the appendix containsa calculation that is referred to in 2.

2 Set-up and statement of results

We now explain the set-up for the analysis and some preliminary results on invariant manifolds.Our main results are precisely stated in 2.3.

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2.1 Equations and scaling variables

The scalar, viscous Burgers equation is the initial value problem

tu = 2xu uux

u|t=0 = h, (2.1)and we assume the viscosity coefficient is small: 0 < 1. For reasons described below, itis convenient to work in so-called similarity or scaling variables, defined as

u(x, t) =1

1 + tw

x

1 + t, log(t + 1)

=x

1 + t, = log(t + 1). (2.2)

In terms of these variables, equation (2.1) becomes

w = Lw www|=0 = h, (2.3)

where Lw = 2w + 12(w).

We will study the evolution of (2.3) in the algebraically weighted Hilbert space

L2(m) =

w L2(R) : w2L2(m) =

(1 + 2)m|w()|2d <

.

It was shown in [GW02] that, in the spaces L2(m) with m > 1/2, the operator L generates astrongly continuous semigroup and its spectrum is given by

(L) =n

2, n N

C : Re() 14

m2

. (2.4)

This is exactly the reason why the similarity variables are so useful. Equation (2.4) shows thatthe operator L has a gap, at least for m > 1/2, between the continuous part of the spectrumand the zero eigenvalue. As m is increased, more isolated eigenvalues are revealed, allowingone to construct the associated invariant manifolds (see below for more details). In contrast,the linear operator in equation (2.1), in terms of the original variable x, has spectrum given by(, 0], which prevents the use of standard methods for constructing invariant manifolds.

For future reference, we remark that the eigenfunctions associated to the isolated eigenvalues = n/2 are given by

0() =14

e2

4 , n() = (n 0)(). (2.5)

See [GW02] for more details. Smoothness and well-posedness of equations (2.1) and (2.3) canbe dealt with using standard methods, for example information about the linear semigroups andnonlinear estimates using variation of constants.

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2.2 Invariant manifolds

We now present the construction, in the phase space of (2.3), of the explicit, global, one-dimensional center manifold that consists of self-similar stationary solutions. We remark thatthis is similar to the global manifold of stationary vortex solutions of the two-dimensional Navier-

Stokes equations, analyzed in [GW05]. First, note that stationary solutions satisfy

(w +1

2w 1

2w2) = 0.

They can be found explicitly by integrating the above equation and rewriting it as

(e2/(4)w)

(e2/(4)w)2=

1

2e

2/(4).

Integrating both sides of this equation from to leads to the following self-similar stationarysolution, for each 0 R:

w() = 0e2/(4)

1 02 e

2/(4)d.

Note that equation (2.3) preserves mass, and we can therefore characterize these solutions byrelating the parameter 0 to the total mass M of the solution. We have

M =

w()d = 0

e2/(4)

1 02

e

2/(4)dd = 2 log(1 0

),

where we have made the change of variables = 1 02 e

2/(4)d. Therefore, we define

AM() = 0e2/(4)

1 02 e

2/(4)d, 0 =

(1 eM/(2)). (2.6)

These solutions are often referred to as diffusion waves [Liu00].

For m > 1/2, the operator L has a spectral gap in L2(m). By applying, for example, the resultsof [CHT97], we can conclude that there exists a local, one-dimensional center manifold near theorigin. In addition, because each member of the family of diffusion waves is a fixed point for(2.3), they must be contained in this center manifold. Thus, this manifold is in fact global, asindicated by figure 1.

Remark 2.1. Another way to identify this family of asymptotic states is by means of the Cole-Hopf transformation, which works for the rescaled form of Burgers equation as well as for theoriginal form (2.1). If w is a solution of (2.3), define

W(, ) = w(, )e12

Rw(y,)dy = 2 exp

1

2

w(y, )dy

. (2.7)

A straightforward computation shows that W satisfies the linear equation

W = LW. (2.8)

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Conversely, let W be a solution of (2.8) for which 1 12 W(y, )dy > 0 for all R and

> 0. Then the inverse of the above Cole-Hopf transformation is

w(, ) = 2 log

1 12

W(y, )dy

. (2.9)

The family of scalar multiples of the zero eigenfunction, 00(), where 0 is given in (2.5), isan invariant manifold (in fact, an invariant subspace) of fixed points for (2.8). Thus, the imageof this family under (2.9) must be an invariant manifold of fixed points for (2.3). Computingthis image leads exactly to the family (2.6), where 0 =

40.

Remark 2.2. One can prove that this self-similar family of diffusion waves is globally stableusing the entropy functional

H[w]() =

R

w(, )e12

R w(y,)dy log

w(, )e

12

R w(y,)dy

e2

4

d.

This is just the standard Entropy functional for the linear equation (2.8) with potential 2/(4),in combination with the Cole-Hopf transformation. For further details regarding these facts, see[DiF03].

We next construct the manifold of diffusive N-waves. Recall that, by (2.4), if m > 3/2 thenboth the eigenvalue at 0 and the eigenvalue at 1/2 are isolated. The latter will lead to a onedimensional stable manifold at each stationary solution.

To see this, define w = AM + v and obtain

v = AM v vvAM v = Lv (AMv), (2.10)

where AM is just the linearization of (2.3) about the diffusion wave with mass M. One cansee explicitly, using the Cole-Hopf transformation, that the operators L and AM are conjugatewith conjugacy operator given explicitly by

AM U = ULU =

e12

RAM(y)dy

, U1 =

e12

RAM(y)dy

. (2.11)

Thus, one can check that the spectra of the operators AM and L are equivalent in L2(m).Furthermore, we can see explicitly that the eigenfunctions of AM are given explicitly by

n() =

n(y)dy

1 2

e

24 d

,

where n is an eigenfunction of L. Notice that, up to a scalar multiple, 1 = AM. If wechose m > 3/2, by (2.4) we can then construct a local, two-dimensional center-stable manifold

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near each diffusion wave. We wish to show that, if the mass is chosen appropriately, then thismanifold is actually one-dimensional. Furthermore, we must show that this manifold is a globalmanifold.

To do this, we appeal to the Cole-Hopf transformation. Using ( 2.7), we define

V(, ) = v(, )e1

2R v(y,)dy

and find that V solves the linear equation

V = AM V.Thus, the two-dimensional center-stable subspace is given by span{0, 1}. The adjoint eigen-function associated with 0 is just a constant. Therefore, if we restrict to initial conditions thatsatisfy

R

V(, 0)d = 0, (2.12)

then the subspace will be one dimensional and given by solutions of the form

V(, ) = 11()e

2 .

One can check that condition (2.12) is equivalent toR

v(, 0)d = 0.

Since w = AM + v, we can insure this condition is satisfied by choosing the diffusion wave thatsatisfies

M = R

AM()d = R

w(, 0)d.

Thus, near each diffusion wave of mass M, there exists a local invariant foliation of solutionswith the same mass M that decay to the diffusion wave at rate e

12.

To extend this to a global foliation, we simply apply the inverse Cole-Hopf transformation (2.9),as in Remark 2.1, to the invariant subspace

{V(, ) = 11()} = {V(, ) = 1AM}.This leads to the global stable invariant foliation consisting of solutions to (2.10) of the form

vN(, ) =1e

2 A

M()

1 12

e2 AM()

.

Using the relationship between v and w, this foliation leads to a family of solutions of (2.3) ofthe form

wN(, ) = AM() + vN(, ) = AM() +1e

2 AM()

1 12e2 AM()

. (2.13)

Below it will be convenient to use a slightly different formulation of this family, which we nowpresent.

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The subspace span{0, 1}, corresponding to the first two eigenfunctions in (2.5), is invariantfor equation (2.8). Therefore, as in Remark 2.1, by using the inverse Cole-Hopf transformation(2.9) we immediately obtain the explicit, two parameter family

wN(, ) =00() + 1e

2 1()

1

0

2 0(y)dy

1

2

e2 0()

, (2.14)

where0 =

40 = 2(1 eM2 ). (2.15)

Based on the above analysis, (2.14) and (2.13) are equivalent. Note that, although the methodused to produce (2.14) is much more direct than that of (2.13), we needed to use the operatorAM and its spectral properties to justify the claim that this family does in fact correspond toan invariant stable foliation of the manifold of diffusion waves.

We now explain why solutions of the form (2.14) are referred to as the family of diffusive N-waves.As mentioned in 1, this terminology was justified in [KT01] by showing that each solution wNis close to an inviscid N-wave pointwise in space. Since we are working in L

2

(m), we need toprove a similar result in that space.

Recall some facts about the N-waves, which can be found, for example, in [ Liu00]. Define

p = 2infy

y

u(x)dx, and q = 2 supy

y

u(x)dx, (2.16)

which are invariant for solutions of equation (2.1) when = 0. (Note that our definitions of pand q differ from those in [KT01] by a factor of 2.) The mass satisfies M = (q p)/2. We willrefer to q as the positive mass of the solution and p as the negative mass of the solution.

The associated N-wave is given by

Np,q(x, t) =

xt+1 if

p(t + 1) < x 0. Using the calculation in the appendix, one can relatethe quantity 1 in (2.14), for any fixed , to the quantities p and q via

1e

2 = 43/2ep/(4) 1

+ O() for 0 < q < p, (2.19)

and a similar result holds for q > p > 0. Two key consequences of this, which will be usedbelow, are

The quantities 0 and 1 are related via01

= exp, (2.20)

The values of p and q for the diffusive N-waves change on a timescale of = O( 1). (Recallthey are only invariant for = 0.)

This second property, which can be seen by differentiating (2.19) with respect to , will lead tothe slow drift along the manifold of diffusive N-waves (see below for more details).

The following lemma, which will be proved in 3, states precisely that there exists an N-wavethat is close in L2(m) to each member of the family wN, at least if the viscosity is sufficientlysmall, thus justifying the terminology diffusive N-wave.

Lemma 2.1. Given any positive constants , p and q, let wN(, ) be a member of the family(2.14) of diffusive N-waves such that, at time = 0, the positive mass of wN(, 0) is q and thenegative mass is p. There exists a 0 > 0 sufficiently small such that, if 0 < < 0, then

wN(, 0) Np.q()L2(m) <

2.3 Statement of main results

We have seen above that the phase space of (2.3) does possess the global invariant manifoldstructure that is indicated in figure 1. To complete the analysis, we must prove our two mainresults, which provide the fast time scale on which solutions approach the family of diffusive N-waves and the slow time scale on which solutions decay, along the metastable family of diffusive

N-waves, to the stationary diffusion wave.

Theorem 1. (The Initial Transient) Fix m > 3/2. Letw(, ) denote the solution to theinitial value problem (2.3) whose initial data has mass M, and letNp,q() be the inviscid N-wavewith values p and q determined by the initial data w(, 0) = h() L2(m). Given any > 0,there exists a T > 0, which is O(| log |), and sufficiently small so that

||w(T) Np,q||L2(m) . (2.21)

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This theorem states that, although the quantities p and q are determined by the initial dataw(, 0), w is close to the associated N-wave, Np,q, at a time = T = O(| log |). The reason forthis is that p = p() and q = q() change on a time scale ofO(1/), which can be seen usingequation (2.19) and is slower than the initial evolution of w. The rate of change of p and q alsodetermines the rate of motion of solutions along the manifold of diffusive N-waves, as illustratedin Figure 1. Note that this theorem states that the solution will be close to an inviscid N-waveafter a time T = O(| log |). By combining this with Lemma 2.1, we see that the solution is alsoclose to a diffusive N-wave.

We remark that the timescale O(| log |) was rather unexpected. We actually expected toapproach a diffusive N-wave on a time scale O(1), although we have not yet been able to obtainthis stronger result. However, these time scales correspond well with the numerical observationsof [KT01, Figure 1], where one can see that, for = 0.01, the solution looks like a diffusiveN-wave at time 2 and a diffusion wave at time 100.

Remark 2.3. To some extent, this fast approach to the manifold of N-waves can be thought ofin terms of the Cole-Hopf transformation (2.7), which depends on. For small, this nonlinear

coordinate change can reduce the variation in the solution for || large. This is illustrated infigure 5.1 of [KN02]. If is small enough, the Cole Hopf transformation can make the initialdata look like an N-wave even before any evolution has taken place.

Theorem 2. (Local Attractivity) There exists a c0 sufficiently small such that, for anysolution w(, ) of the viscous Burgers equation (2.3) for which the initial conditions satisfy

w|=0 = w0N + 0,where w0N is a diffusive N-wave and 0L2(m) c0, there exists a constant C such that

w(, ) = wN(, ) + (, ),with wN the corresponding diffusive N-wave solution and

(, )L2(m) Ce.

By combining these results, we obtain a geometric description of metastability. Theorem 1 andLemma 2.1 tell us that, for any solution, there exists a T = O(| log |) at which point the solutionis near a diffusive N-wave. By using Theorem 2 with this solution at time T as the initialdata, we see that the solution must remain near the family of diffusive N-waves for all time. As

remarked above, the time scale ofO(1/) on which the solution decays to the stationary diffusionwave is then determined by the rates of change of p() and q() within the family of diffusiveN-waves. In other words, near the manifold of diffusive N-waves, w(, ) = wN(, ) + (, ),where (, ) e and wN(, ) is approaching a self-similar diffusion wave on a timescaledetermined by the rates of change of p() and q(), which are O(1/).

We remark that it is not the spectrum of L that determines, with respect to , the rate ofmetastable motion. Instead, this is given by the sizes of the coefficients 0 and 1 in the spectralexpansion and their relationship to the quantities p and q.

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3 Proof of Lemma 2.1

We now prove Lemma 2.1.

In [KT01], Kim and Tzavaras prove that the inviscid N-wave is the point-wise limit, as 0,of the diffusive N-wave. Here we extend their argument to show that one also has convergencein the L2(m) norm. For simplicity we will check explicitly the case in which 0 < q < p - the casein which q is larger than p follows in an analogous fashion. However, we note that we do requirethat both p and q be nonzero which is why we stated in the introduction that our results holdonly for almost all initial conditions. See Remark 3.1, below.

Using equation (2.13) we can write the diffusive N-wave with positive and negative mass givenby q and p at time 0 as

wN(, 0) =00() + 1e

0/21()

1 02

0(y)dy 12e0/20()=

00() + 11()

1 02

0(y)dy 120()where for notational simplicity we have defined 1 = 1e

0/2. If we now recall that 1() =

20(), we can rewrite the expression for the wN as

wN(, 0) = 20

1

1 210()

{1 02

0(y)dy}

. (3.22)

We need to prove that

(1 + 2)m(wN(, 0) Np,q())2 < 2 .

Well give the details for 0

(1 + 2)m(wN(, 0) Np,q())2 < 2/2 .

The integral over the negative half axis is entirely analogous.

Break the integral over the positive axis into three pieces - the integral from [0 ,

q ], theintegral from [

q , q + ] and the integral from [q + , ). Here is a small constant that

will be fixed in the discussion below. We refer to the integrals over each of these subintervals asI, II, and III respectively and bound each of them in turn.

The simplest one to bound is the integral II. Note that, using equations (2.15) and (2.19), thedenominator in (3.22) can be bounded from below by 1/2 and, thus, the integrand is can bebounded by C(1 + (

q + )2)mq. Therefore, if

q, Np,q() = 0 so

III =

q+

(1 + 2)m(wp,q(, 0))2d (3.23)

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To estimate this term we begin by considering the denominator in (3.22). Using (2.18) and(2.19), we have

1 02

0(y)dy = e14

(pq) +02

0(y)dy. (3.24)

Thus, the full denominator in (3.22) has the form

1 210()

e

14

(pq) +02

0(y)dy

= 1 4

3/2

1e

14

(pq)e2/(4) +

0

1

0(y)dy

0()

= 1 +4

3/2

1

3/2ep/(4) + O() e14

(pq)e2/(4) + exp

= 1 +

e

14

(2q)

1 + O(1ep/(4))

+ exp,

where exp denotes terms that are exponentially small in (i.e. contain terms of the formep/(4) or eq/(4)), uniformly in . Note that, in the above, the term

0(y)dy/0() was

bounded uniformly in using the estimatex

es2

2 ds 1x

ex2

2 , for x > 0,

which can be found in [KS91, Problem 9.22]. But with this estimate on the denominator of wN,we can bound the integral III by

III Cq+

(1 + 2)m( 201

)2(1 + e14

(2q))2d,

where the constant C can be chosen independent of for < 0 if0 is sufficiently small. Thisintegral can now be bounded by elementary estimates, and we find

III Ceq/2,where the constant C depends on q but can be chosen independent of .

Finally, we bound the integral I. Note that for 0 < 0, then there exists a sufficiently small and a T sufficientlylarge (O(| log |) as 0) such that

w(, T) Np,q()L2(m) < .

We estimate the norm by breaking the corresponding integral into subintegrals using (, p), (p , p + ), (p + , ), (, ), (, q ), (q , q + ) and (q + , ).Note that the integrals over the short intervals can all be bounded by C, so well ignorethem. Well estimate the integrals over (,

q ) and (q + , ) - the remaining two are very

similar.

First, consider the region where >

q + . In this region, N() 0, so we only need to showthat, given any > 0, there exists a sufficiently small and T > 0, ofO(| log |), such that

q+

(1 + 2)m|w(, )|2d < .

Consider the formula for w, given in equation (4.26). To bound this, we must bound thedenominator from below and the numerator from above. We will first focus on the denominator.

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Combining the estimates for the Jis, equations (4.32) - (4.34), and the estimate for the denom-inator (4.31), we have

q+

(1 + 2)m|w(, )|2d Cq+

(1 + 2)meM e

2C(R,||h||m)e

2

2 d

+ Cq+

(1 + 2)m1

e1

C(R,

||h||m)

(Re

2 )2( + Re

2 )2e1

2[q(Re

/2)2]

d

+ C

q+

(1 + 2)me1C(R,||h||m)e

12

[q(Re/2)2]d

+ C

q+

(1 + 2)m1

e1C(R,||h||m)|

Re/2

e14

()2h(e/2)e+12

(MH(e 2 ))d|2d

I + II + III+ IV.

We now estimate term II. Terms I and III are similar. Define z = q (0, ). Recallingthat has been chosen sufficiently large so that Re

2 < /2, we have

e12

[q(Re/2)2] e 12z2e 18 2e 12 q.

Therefore, we have

|II| C

e1C(R,||h||m)

0

(1 + (z +

q + )2)m(z +

q +3

2)e

12z2e

182e

12qdz

C(q)2m+1e 12 q 1

e1C(R,||h||m)e

182.

This can be made as small as we like (for any q) by choosing R large enough so that C(R, ||h||m) 0. Finally, we have

|I3| eM2q4 2Re/2e

14

(Re/2)2 .

For the numerator, we have

|J1

| eC(R,||h||m)

2 Re/2

e

14

()2d

eC(R,||h||m)2+Re/2

(z )e z2

4dz

CeC(R,||h||m)2 e (+Re/2)24 + Ce

(+Re/2)24

Ce 2

4 .

where we have used the fact that > 0. Next,

|J2

| e

M2 e

C(R,||h||m)2 4

Re/24

(

4z)ez

2

dz

CeM2 eC(R,||h||m)2 e (Re/2)24 .

Lastly,

|J3| CeM2 eC(R,||h||m)

2 (Re/2)2e14

(Re/2)2

17


18/22

Therefore, we haveq

(1 + 2)m|w(, ) N()|2d

Cq

(1 +

2

)

meM2e

C(R,||h||m)2 e

(Re/2)24

e 12 (M+C(R,||h||m)) 2

d

C1

q

(1 + 2)m+1e2C(R,||h||m)e

(Re/2)22 d

C1

e2C(R,||h||m)

q20

(1 + (z + )2)me2

8 ez2 e

z2

2 dz

C 1

e2C(R,||h||m)e

2

8 ,

where we have used the change of variables = z + . This can be made small by first choosing

R large enough so that 2C(R, ||h||m) < 2

/8, and then taking small.

5 Proof of Theorem 2

In the previous section we saw that in a time = O(| log |) we end up in an arbitrarily small(but O(1) with respect to ) neighborhood of the manifold of diffusive N-waves. In the presentsection we show that this manifold is locally attractive by proving Theorem 2.

Remark 5.1. An additional consequence of the proof of this theorem is that the manifold of

diffusive N-waves is attracting in a Lyapunov sense because the rate of approach to it, O(e),is faster than the decay along it, O(e/2). Note that this is not immediate from just spectralconsiderations since this manifold does not consist of fixed points. Therefore, the eigendirectionsat each point on the manifold can change as the solution moves along it.

Remark 5.2. In [KN02, 5], the authors make a numerical study of the metastable asymptoticsof Burgers equation. Their numerics indicate that, while the rate of convergence toward thediffusive N-wave (e in our formulation) seems to be optimal, the constant in front of theconvergence rate (C in our formulation) can be very large for some initial conditions. In fact,our proof indicates that this constant can be as large as O(1/)max{1, eM/2}.

To prove the theorem note that by the Cole-Hopf transformation we know that if w(, ) =wN(, ) + (, ) solves the rescaled Burgers equation, where wN is given in equation (2.14),then

W(, ) = (wN(, ) + (, ))e 1

2

R(wN(y,)+(y,))dy

is a solution of the (rescaled) heat equation:

W = LW. (5.35)

18


19/22

We now write W = VN+, where VN = wN exp( 12 wN(y, )dy) = 00()+1e

/21().That is, VN is the heat equation representation of the diffusive N-wave, which we know is alinear combination of the Gaussian, 0, and 1.

With the aid of the Cole-Hopf transformation we can show that decreases with the rateclaimed in the Proposition. To see this, first note that if we choose the coefficients 0 and 1

appropriately we can insure that(, 0) =

(, 0)d = 0.

This follows from the fact that the adjoint eigenfunctions corresponding to the eigenfunctions0 and 1, respectively, are just 1 and . This in turn means that there exists a constant Csuch that

(, )L2(m) Ce, (5.36)at least if m > 5/2. Integrating the Cole-Hopf transformation we find

(VN(y, ) + (y, ))dy = 2e 12 R((wN(y,)+(y,))dy 1 (5.37)and, in the case = 0,

VN = 2

e

12

RWN 1

. For later use we note the following

easy consequence of (5.37):

Lemma 5.1. There exists a constant N > 0 such that for all 0 we have

1

1

2(VN(y, ) + (y, )) dy = e

12

R(wN(y,)+(y,))dy N .

Proof. For any finite the estimate follows immediately because of the exponential. The onlything we have to check is that the right hand side does not tend to zero as . Howeverthis follows from the fact that we know (from a Lyapunov function argument, for example) thatwN(, ) + (, ) AM() as , where AM is one of the self-similar solutions constructedin Section 1 and hence

e12

R(wN(y,)+(y,))dy e 12

RAM(y) = 1(1eM/2)

0(y)dy min{1, eM/2} > 0 .

Next note that by rearranging (5.37) we find

(y, )dy = 2 log

1 12

(VN(y, ) + (y, )) dy

1 12 VN(y, )dy

. (5.38)

Differentiating, we obtain the corresponding formula for , namely,

(, ) = 12

(, ) VN(y, )dy VN(, )

(y, )dy 2(, )

1 12(VN(y, ) + (y, ))dy

1 12

VN(y, )dy

(5.39)19


20/22

But now, by Lemma 5.1 the denominator of the expression for can be bounded from below by2N, while in the numerator

VN and VN are bounded in time while

and are each

bounded by C,e which leads to the bound asserted in Theorem 2.

6 Concluding remarks

In the above analysis, the global, one-dimensional manifold of fixed points governing the long-time asymptotics is constructed directly, in a manner similar to that of [GW05] for the Navier-Stokes equations in two-dimensions. However, we then utilized the Cole-Hopf transformation toextend the stable foliation of this manifold to a global foliation, which consists of the diffusiveN-waves that govern the metastable behavior. Ultimately one would like to obtain a similargeometric description of, for example, the numerically observed metastability in [YMC03] forthe two-dimensional Navier-Stokes equations. In order to do this one would need an alternativeway to construct a global foliation of the manifold of fixed points. This would be an interesting

direction for future work.

7 Acknowledgements

The authors wish to thank Govind Menon for bringing to their attention the paper [KT01], whichlead to their interest in this problem. The second author also wishes to thank Andy Bernoff forinteresting discussions and suggestions about this work. Margaret Beck was supported in partby NSF grant number DMS-0602891. The work of C. Eugene Wayne was supported in part by

NSF grant number DMS-0405724. Any findings, conclusions, opinions, or recommendations arethose of the authors, and do not necessarily reflect the views of the NSF.

8 Appendix

We now give the calculation that leads to (2.19). Using the definition of p in (2.16), we findthat

p =

2infyy

wN

(, )d = 4 supyy

log 1 0

2

0(y)dy

1

2e

2

0() d

= 4 supy

log

1 0

2

y

0(z)dz 12

e2 0(y)

.

A direct calculation shows that the supremum is achieved at

y =20

1e

2

.

20


21/22

Substituting in this value and rearranging terms, we find

12

e2 0(y

) = ep4

1 0

2

y

0(z)dz

.

Since y 0(z)dz (0, 1), the value of 0 in (2.15) implies that the right hand side satisfiese

p4 1 e p4

1 0

2

y

0(z)dz

e p4 eM2

if M > 0, and

ep4 eM2 e p4

1 0

2

y

0(z)dz

e p4 1

if M < 0. Because 0(y) 0 and p 2|M|, we see that, if is sufficiently small, then 1 0.Furthermore, 1 = 0 if p = M/2, ie M < 0 and q = 0. Using the fact that (y) 1/4,we see that

1e 2 24 e p4 1 c 02 ,where c (0, 1). This leads to the estimate (2.19), at least when both q = 0 and p = 0.

We remark that 1 0 and the fact that

1 02

y

0(z)dz = 1 (1 eM2 )y

0(z)dz

(eM2 , 1) if M > 0

(1, eM2 ) if M < 0

implies that the denominator in the definition of wN (2.14) is never zero.

References

[BKS95] H. Berestycki, S. Kamin, and G. Sivashinsky. Nonlinear dynamics and metastabilityin a Burgers type equation (for upward propagating flames). C. R. Acad. Sci. ParisSer. I Math., 321(2):185190, 1995.

[CHT97] X. Chen, J. K. Hale, and B. Tan. Invariant foliations for c1 semigroups in banachspaces. Journal of Differential Equations, 139:283318, 1997.

[CP89] J. Carr and R. L. Pego. Metastable patterns in solutions ofut = 2uxxf(u). Comm.Pure Appl. Math., 42(5):523576, 1989.

[CP90] J. Carr and R. L. Pego. Invariant manifolds for metastable patterns in ut = 2uxx

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[Daf05] C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math-ematical Sciences]. Springer-Verlag, Berlin, second edition, 2005.

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[DiF03] M. DiFrancesco. Diffusive behavior and asymptotic self similarity for fluid models.PhD thesis, University of Rome, 2003.

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[GW02] T. Gallay and C. E. Wayne. Invariant manifolds and the long-time asymptotics of thenavier-stokes and vorticity equations on R2. Arch. Rational Mech. Anal., 163:209258,2002.

[GW05] T. Gallay and C. E. Wayne. Global stability of vortex solutions of the two-dimensionalNavier-Stokes equation. Comm. Math. Phys., 255(1):97129, 2005.

[KN02] Y.-J. Kim and W.-M. Ni. On the rate of convergence and asymptotic profile ofsolutions to the viscous Burgers equation. Indiana Univ. Math. J., 51(3):727752,

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[KS91] I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus, volume 113of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.

[KT01] Y.-J. Kim and A. E. Tzavaras. Diffusive N-waves and metastability in the Burgersequation. SIAM J. Math. Anal., 33(3):607633 (electronic), 2001.

[Liu00] T.-P. Liu. Hyperbolic and Viscous Conservation Laws. Number 72 in CBMS-NSFRegional Conference Series in Applied Mathematics. SIAM, 2000.

[SW99] X. Sun and M. J. Ward. Metastability for a generalized Burgers equation with appli-cations to propagating flame fronts. European J. Appl. Math., 10(1):2753, 1999.

[Whi99] G. B. Whitham. Linear and nonlinear waves. Pure and Applied Mathematics (NewYork). John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original, AWiley-Interscience Publication.

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