Gui-Qiang Chen, Marshall Slemrod and Dehua Wang- Vanishing Viscosity Method for Transonic Flow

8/3/2019 Gui-Qiang Chen, Marshall Slemrod and Dehua Wang- Vanishing Viscosity Method for Transonic Flow

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VANISHING VISCOSITY METHOD FOR TRANSONIC FLOW

GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG

Dedicated to Constantine M. Dafermos on the Occasion of His 65th Birthday

Abstract. A vanishing viscosity method is formulated for two-dimensional transonicsteady irrotational compressible fluid flows with adiabatic constant [1, 3). Thisformulation allows a family of invariant regions in the phase plane for the correspondingviscous problem, which implies an upper bound uniformly away from cavitation for theviscous approximate velocity fields. Mathematical entropy pairs are constructed throughthe Loewner-Morawetz relation by entropy generators governed by a generalized Tricomiequation of mixed elliptic-hyperbolic type, and the corresponding entropy dissipationmeasures are analyzed so that the viscous approximate solutions satisfy the compensatedcompactness framework. Then the method of compensated compactness is applied toshow that a sequence of solutions to the artificial viscous problem, staying uniformlyaway from stagnation, converges to an entropy solution of the inviscid transonic flowproblem.

1. Introduction

In two significant papers written a decade apart, Morawetz [27, 28] presented a programfor proving the existence of weak solutions to the equations governing two-dimensionalsteady irrotational inviscid compressible flow in a channel or exterior to an airfoil. As is

well known, the classical results of Shiffman [35] and Bers [4] apply when the upstreamspeed is sufficiently small, for which the flow remains subsonic and the governing equationsare elliptic (also see [12, 14, 15, 16, 17, 18, 19, 20]). However, beyond a certain speed atinfinity (determined by the flow geometry), the flow becomes transonic which, coupledwith nonlinearity, yields shock formation (cf. [26]). Morawetzs program in [27, 28] wasto imbed the problem within an assumed viscous framework for which the compensatedcompactness framework (see Section 6 of this paper) would be satisfied. Under this as-sumption, Morawetz proved that solutions of the as yet unidentified viscous problem havea convergent subsequence whose limit is a solution of the transonic flow problem.

The purpose of this paper is to present such a viscous formulation, hence completingpart but not all of Morawetzs program. Specifically, a vanishing viscosity method is for-mulated for two-dimensional transonic steady irrotational compressible fluid flows with

adiabatic constant [1, 3) to ensure a family of invariant regions for the corresponding

Date: September 22, 2006.1991 Mathematics Subject Classification. Primary: 35M10,76H05, 35A35,76N10,76L05; Secondary:

35D05,76G25.Key words and phrases. Transonic flow, viscosity method, Euler equations, gas dynamics, Bernoullis

law, irrotational, approximate solutions, entropy, invariant regions, compensated compactness framework,convergence, entropy solutions.

1


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2 G.-Q. CHEN, M. SLEMROD, AND D. WANG

viscous problem, which implies an upper bound uniformly away from cavitation for the vis-cous approximate velocity fields. Mathematical entropy pairs are constructed through theLoewner-Morawetz relation by entropy generators governed by a generalized Tricomi equa-

tion of mixed elliptic-hyperbolic type, and the corresponding entropy dissipation measuresare analyzed so that the viscous approximate solutions satisfy the compensated compact-ness framework. Then the method of compensated compactness is applied to show thata sequence of solutions to the viscous problem, staying uniformly away from stagnation,converges to an entropy solution of the inviscid transonic flow problem.

On the other hand, Morawetzs assumption of no stagnation points for the flow has notbeen removed; and the slip boundary condition (u, v)n = 0 on the obstacle is only satisfiedas the inequality (u, v) n 0 for the general case (the desired condition (u, v) n = 0may be achieved when a solution has no jump with certain regularity along the obstacle),where (u, v) is the fluid velocity field and n is the unit normal on the obstacle pointinginto the flow. Both issues may reflect rather complicated boundary layer behavior fortransonic flow [33] and require further investigation, which are the topics for subsequent

research. In addition, we note that, when 3, cavitation is indeed possible and will bethe topic of a sequel to this paper.

This paper is divided into nine sections after the introduction. In Section 2, the classicalfluid equations are presented for isentropic and isothermal irrotational planar flow. InSection 3, the fluid equations are rewritten in the polar variables (u, v) = (q cos , q sin ).In Section 4, we analyzes the behavior of the Riemann invariants in the supersonic region,which guides the design of artificial viscous terms in our vanishing viscosity method. InSection 5, we continue our analysis and set boundary conditions for the viscous system.This yields a family of invariant regions in the (u, v) fluid phase plane for [1, 3), where is the ratio of specific heats if > 1 and = 1 denotes the case of constant temperature.In particular, any invariant region is uniformly bounded away from the cavitation circlein the phase plane, which yields a upper bound uniformly away from cavitation in the

viscous approximate solutions. In Section 6, we formulate a compensated compactnessframework for steady flow. In Section 7, we first construct all mathematical entropypairs for the potential flow system through the Loewner-Morawetz relation by entropygenerators governed by the generalized Tricomi equation of mixed elliptic-hyperbolic type.Then we introduce the notion of entropy solutions through an entropy pair by a convexentropy generator suggested by the work of Osher-Hafez-Whitlow [32]. In Section 8, we usethe previously derived uniform L bounds on (u, v), plus the convex entropy generatorintroduced in Section 7 to guarantee our problem is within the compensated compactnessframework. In Section 9, we develop Morawetzs argument in [28] for proving convergenceof a subsequence of solutions of our viscous problem to the irrotational fluid problem.Finally, in Section 10, we prove the existence of smooth solutions to our viscous problem.

2. Mathematical Equations

Consider the artificial viscous system in a domain R2:vx uy = R1,(u)x + (v)y = R2,

(2.1)


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VANISHING VISCOSITY METHOD FOR TRANSONIC FLOW 3

where R1 and R2 are the artificial viscosity terms to be determined, and (u, v) is the flowvelocity field. For a polytropic gas with the adiabatic exponent > 1, p = p() = / isthe normalized pressure, the renormalized density is given by Bernoullis law:

= (q) =

1 12

q2

11 , (2.2)

where q is the flow speed defined by q2 = u2 + v2. The sound speed c is defined as

c2 = p() = 1 12

q2. (2.3)

At the cavitation point = 0,

q = qcav :=

2

1 .

At the stagnation point q = 0, the density reaches its maximum = 1. Bernoullis law(2.2) is valid for 0 q qcav. At the sonic point q = c, (2.3) implies q2 = 2+1 . Definethe critical speed qcr as

qcr :=

2

+ 1.

We rewrite Bernoullis law (2.2) in the form

q2 q2cr =2

+ 1

q2 c2 . (2.4)

Thus the flow is subsonic when q < qcr, sonic when q = qcr, and supersonic when q > qcr.For the isothermal flow ( = 1), p = c2 where c > 0 is the constant sound speed, and

the density is given by Bernoullis law:

= 0 exp u2 + v2

2c2 (2.5)

for some constant 0 > 0. In this case, qcr = c. After scaling, we can take c = 0 = 1.

3. Formulation in Polar Coordinate Phase Plane

We now use the polar coordinates in the phase plane:

u = q cos , v = q sin ,

and rewrite the viscous conservation laws (2.1) in terms of (q, ). Write the second equationof (2.1) as

xu + ux + yv + vy = R2

or

(q)qxu + ux + (q)qyv + vy = R2,and use

q =

u2 + v2, qx =1

q(uux + vvx), qy =

1

q(uuy + vvy),

(q) = qc2

to find

(c2 u2)ux uv(vx + uy) + (c2 v2)vy = c2

R2.


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Then (2.1) becomes

Auv

x

+ B uv

y

= R1c2

R2 ,where

A =

0 1

c2 u2 uv

, B =

1 0

uv c2 v2

.

Thus, in terms of (q, ), we obtain

A1

q

x

+ B1

q

y

=

R11q

R2

, (3.1)

where

A1 = sin q cos c2q2

c2qcos sin

, B1 =

cos q sin c2q2c2q

sin cos

.

In terms of (, ), using

x = qqx = qc2

qx, y = qc2

qy,

and thus

q

x

= c2q

xx ,

q

y

= c2q

yy ,

we find

A2

x

+ B2

y

=

R1R2

, (3.2)

where

A2 =

c2

q sin q cos 1q

(c2 q2)cos q sin

, B2 =

c2q cos q sin

1q

(c2 q2)sin q cos

.

We symmetrize (3.2) to obtain

A3

x

+ B3

y

= R1

q2

c2q2 R2

, (3.3)

where

A3 =

c2

qsin q cos

q cos q3 sin c2q2

, B3 =

c2q

cos q sin q sin q3 cos

c2q2

.


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4. Choice of Artificial Viscosity from an Analysis of Riemann Invariants

The two matrices A1, B1 in (3.1) commute, thus their transposes commute and theyhave common eigenvectors. The eigenvalues of A1 and B1 are

= sin

q2 c2c

cos , = cos

q2 c2c

sin .

The left eigenvectors of A1 are

(

q2 c2qc

, 1),

and thus the Riemann invariants W satisfy

W

= 1,W

q=

q2 c2

qcfor q c. (4.1)

Multiply (3.1) by (Wq ,W ) to obtain

Wx

+ Wy

= Wq

R1 + 1q

W

R2. (4.2)

We now consider the viscosity terms of the form:

R1 = (1(, )) , R2 = (2(, )) . (4.3)In particular, for a special choice of 1 and 2, we have the desired relations for theRiemann invariants.

Proposition 4.1. If

1 = 1, 2 = 1 c2

q2for q > c, (4.4)

then the Riemann invariants W

satisfy the following equations:

qcq2 c2

Wx

+ W

y

W = c( 3)q

2 + 4c2

22q2

q2 c2||2 (4.5)

for 1.Proof. We first focus on the + part, since the - part can be done analogously. Theproof is divided into three steps.

Step 1: Substitute (4.1) into (4.2) to obtain

+W+

x+ +

W+y

=

q2 c2

qcR1 +

1

qR2. (4.6)

Multiplication of (4.6) by qc

q2

c2

gives

qcq2 c2

+

W+x

+ +W+

y

=

W+

R1 +c

q2 c2 R2.

Using ddq

= q/c2, we haveW+

=

W+q

dq

d=

c

q2 c2q2

,


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From the definition of W, we have

Wq

= q2 1

q.

Using the substitution q = sec t, we can integrate to find

W =

q2 1 arccos(q1) + C =

q2 1 1

arccos(q1) 4

,

where the constants C is set to be C = (

q2 1 arccos(q1))|q=2 = (1 4 ).Along the level curves W = const., we have

d

dq=

q2 1

q.

Thus, on the level set of W+,ddq

> 0 and is increasing when q is increasing; while, on

the level set of W, ddq < 0 and is decreasing when q is increasing.

If (x0, y0) = 0, q(x0, y0) = 2 and we leave q = 2, we have(x, y) 0

q2 1 1

arccos(q1) 4

(stay below W+),

(x, y) 0

q2 1 1

+

arccos(q1) 4

(stay above W),

that is,q2 1 1

arccos(q1) 4

|(x, y)0| inside the apple shaped region.

See Figure 1 with qcav = when = 1.The same situation occurs when the level set curves W = W(q0, 0) for (x0, y0) = 0

and q(x0, y0) = q0 >

2, for which we obtain similar invariant regions of apple shape

past the point (q0, 0) in the (u, v)-plane.Now we return to the issue of boundary conditions. Denote the boundary of the bounded

domain by , the boundary of the obstacle by 1, and the far field boundary by 2.Thus, = 1 2. Since we do not want W+ to have a minimum on boundary 1and W to have a maximum on boundary 1, we require

W+n

< 0,W

n> 0 at all boundary points,

where n is the unit normal into the flow region on . Recall

W = q2 1

qq.

Therefore, if we set n = 0 on 1,

thensign(W n) = sign(q n)

at those boundary points where q > 1, i.e. where the Riemann invariants are defined.Hence, one resolution of the boundary condition issue is to set

2 n = | (u, v) n| on 1,


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0

W+ = 0

q =2qcr

q = qcr

0

W= 0

q = qcav

u

v

Figure 1: Invariant regions of apple shape.

and

(u, v) (u, v) = 0 on 2,with q = |(u, v)| (0, qcav) = (0,). Trivially, W are not even defined on 2 andhence cannot have a maximum or minimum there. Since

= q ddq

= qq,

we have automatically

sign(W n) = sign( n) = 1 on all boundaries,and our minimum principle for W+ and maximum principle for W are indeed valid.

Secondly, they formally yield

(u, v) n = 0 on 1,and

(u, v) (u, v) 0 as x2 + y2 ,if 0. This is the case when any shock strength is zero at its intersection point with theboundary as conjectured for the shock formed in the supersonic bubble near the obstacle


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(see Morawetz [29]). Finally, we see that the relation b etween 1 and 2 is only necessaryfor q 2. Any smooth continuations of1 and 2 inside q =

2 with 1, 2 > 0 suffices.

With this, we conclude that (q, ) stay inside the apple region for = 1. More

generally, for 1 < < 3, we haveTheorem 5.1. Consider the viscous problem

vx uy = (1()) ,(u)x + (v)y = (2()) ,

(5.1)

with

1 = 1, 2 =q2 c2

q2for q

2qcr, 1 < 3,

and the boundary conditions:

n = 0 on 1,2 n = | (u, v) n| on 1,(u, v) (u, v) = 0 on 2 with q < qcav,

(5.2)

Then all solutions to (5.1)(5.2) are bounded on the domain staying uniformly in away from cavitation, i.e. there exists q < qcav such that q q that is equivalent to(q) > 0 for some = (q) > 0. More specifically, when 1 < 3, the viscousapproximate solutions stay in a family of apple shaped invariant regions as shown in

Figures 1 and 35.

Proof. The theorem has been proved in the above for the isothermal case = 1, c = 1,and qcr = 1.

For the case 1 < < 3, from (4.5), we first require

( 3)q2

+ 4c2

< 0,which implies q2 > 43c

2. Then, from Bernoullis law (2.4), we have

q2 q2cr >2

3 c2 =

2

3

1 12

q2

,

thus q2 > 2q2cr, i.e. q >

2qcr.

We first consider the case that the far field speed q

2qcr. Then, from the definitionof the Riemann invariants (4.1), we follow Landau-Lifshitz [22], page 446, and have

W =

W(q) W(

2qcr)

,

where

W(q) =

+ 1

1 arcsin

12

q2q2cr

1 arcsin

+ 1

2

1 q

2cr

q2

.

The two level set curves of W intersect to form a invariant region of apple shapepast (

2qcr, 0) as in Figure 1 if

W(qcav) W(

2qcr) > ,


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1.5 2 2.5 3

Figure 2: The graph of a()

0

W+ = 0

q = qcr

u

v

W= 0

q = qcav

q =2q

cr

0

(2qcr, 0)

Figure 3: Round apple shaped region (

2qcr, 0) when a() <

that is,

a() := W(qcav)W(

2qcr)

=+ 1

1 1

2+ 1

1 arcsin

12

arcsin

+ 1

4

> .


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The graph of the function a() is shown in Figure 2, which shows that a() = forsome 1.224 and a() > for 1 < .

0

W+ = 0

q =2qcr

q = qcr

0

u

v

W= 0

W+ = + 0

W= + 0

q = qcav

(2qcr, 0) (

2qcr, + 0)

Figure 4: Invariant regions (

2qcr, 0) (

2qcr, + 0) when a() (2 , )

For general [, 3), the level set curves of W = 0 past (

2qcr, 0) end at some

points on the cavitation circle q = qcav and do not intersect each other. We denote(2qcr, 0) the round apple shaped region formed by W+ > 0, W < 0, and thecavitation circle q = qcav; see Figure 3.

When a() (2 , ], we find that, for any 0 [0, 2),(

2qcr, 0) (

2qcr, + 0)

forms an invariant region for the viscous solutions; this follows from application of ourminimum and maximum principle for W+ and W, respectively, at the crossing point onthe new level set curves. Such invariant regions stay away from the cavitation circle (seeFigure 4).

When a() (4 , 2 ], then, for any 0 [0, 2),

(2qcr, 0) (2qcr, 2

+ 0) (2qcr, + 0) (2qcr, 32

+ 0),

forms an invariant region for the viscous solutions, staying away from cavitation; see Figure5.

In general, when a() ( 2n , 2n1 ), n = 1, 2, . . . , for each 0 [0, 2), it requires anintersection of at least 2n round apple shaped regions including (

2qcr, 0) to form an

invariant region staying away from the cavitation circle.


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0

q =2q

cr

q = qcr

0

u

v

q = qcav

Figure 5: Invariant regions (

2qcr, 0)(

2qcr,2 +0)(

2qcr, +0)(

2qcr,

32 +

0) when a() (4 , 2 ]

When the far field speed q

(

2qcr, qcav), then the level set curves of W

=

W(q0, 0), for some q0 (q, qcav) and 0 = arctan vu, past the point (q0, 0) eitherintersects each other or end at some points on the cavitation circle q = qcav. As before, wedenote (q0, 0) the apple shaped region formed by W+ > W+(q0, 0), W < W(q0, 0),and the cavitation circle q = qcav similar to either Figure 1 or Figure 3. Then we can sim-ilarly obtain the invariant regions which consist of either a single apple shaped regionor some unions of certain number round apple shaped regions including (q0, 0), whichstay away from the cavitation but include the state (u, v) (cf. Figures 56).

6. Compensated Compactness Framework for Steady Flow

Let a sequence of functions w(x, y) = (u, v)(x, y), defined on open subset R2,satisfy the following Set of Conditions (A):

(A.1) q(x, y) = |w(x, y)| q a.e. in , for some positive constant q < qcav < ;(A.2) xQ1(w)+ yQ2(w) are confined in a compact set in H1loc (), for any entropy-

entropy flux pairs (Q1, Q2) so that (Q1(w), Q2(w)) are confined in a bounded setuniformly in Lloc().

Then, by the Div-Curl Lemma of Tartar [36] and Murat [30] and the Young measurerepresentation theorem for a uniformly bounded sequence of functions (cf. Tartar [36];


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also Ball [1]), we have the following commutation identity:

< (w), Q1+(w)Q2(w) Q1(w)Q2+(w) >=< (w), Q1+(w) >< (w), Q2

(w) >

< (w), Q1

(w) >< (w), Q2+(w) >,

(6.1)

where = x,y(w), w = (u, v), is the associated family of Young measures (probabilitymeasures) for the sequence w(x, y) = (u, v)(x, y). This is equivalent to

< (w) (w), I(w, w) >= 0, (6.2)where (w) (w) is a product measure for (w, w) R2 R2 andI(w, w) = (Q1+(w)Q1+(w))(Q2(w)Q2(w))(Q2+(w)Q2+(w))(Q1(w)Q1(w)).

The main point for the compensated compactness framework is to prove that isin fact a Dirac measure by using entropy pairs, which implies the compactness of thesequence w(x, y) = (u, v)(x, y) in L1loc(). Some additional references on compensatedcompactness method include Chen [5, 6], Dafermos [10], DiPerna [11], Evans [13], andSerre [34]. Theorem 5.1 shows that (A.1) is satisfied for our viscous solution sequencew = (u, v). In the next two sections, we show how (A.2) can be achieved.

7. Entropy Pairs via Entropy Generators

We first construct all mathematical entropy pairs for the potential flow system.For some function V(, ) to be determined, multiply (3.3) from left by (V, V) to get

the entropy equality

Q1x + Q2y = VR1 + q2

c2 q2 VR2, (7.1)where (Q1, Q2) is defined by

Q1

=c2

qsin V q cos V, Q1

= q cos V q

3 sin

c2

q2

V,

Q2

= c2

qcos V q sin V, Q2

= q sin V + q

3 cos

c2 q2 V.(7.2)

We note that, the appearance of the term c2 q2 in the denominator of (7.2) is onlya consequence of our formalism. It will cancel out as we proceed. Using

2Qi =

2Qi ,

i = 1, 2, from (7.2), we see that V satisfies the Tricomi type equation of mixed type:

c2

qV + qV +

q3

c2 q2 V

= 0. (7.3)

Thus, we have defined an entropy pair (Q1, Q2) generated by V. Alternatively, we candefine an entropy pair (Q1, Q2) generated by H, where H and V are related by

H H = V, H + 1

H = q2

c2 q2 V, (7.4)

with = () defined by () = c2/q2, and H is determined by the generalized Tricomiequation:

H +1

2(1 M2)H = 0, (7.5)

and M = q/c is the Mach number.


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Lemma 7.1. The entropy pairs (Q1, Q2) are given by the Loewner-Morawetz relation:

Q1 = qH cos qH sin , Q2 = qH sin + qH cos , (7.6)

where the generators H are all solutions of (7.5).

This can be seen by differentiation of (7.6) with respect to (, ) and comparison with(7.2).

A prototype of the generators H, as suggested in [32], is

H =2

2+

12

(M2 1)dd, (7.7)

which is a trivial solution of the generalized Tricomi equation (7.5). Notice that H isstrictly convex in (, ) in the supersonic region. We denote by (Q1, Q

2) the corresponding

entropy pair generated by the convex generator H. With this, we introduce the notion

of entropy solutions.Definition 7.1 (Notion of Entropy Solutions). A bounded, measurable vector functionw(x, y) := (u, v)(x, y), q(x, y) qcav, is called an entropy solution of the potential flowsystem in a domain :

vx uy = 0,(u)x + (v)y = 0,

(7.8)

if w(x, y) satisfies (7.8) and the following entropy inequality:

Q1x + Q2y 0 (7.9)

in the sense of distributions in , in addition to the corresponding boundary conditions

in the trace or asymptotic sense on .The physical correctness of the entropy inequality (7.9) is provided by Theorem 2.1 of

Osher-Hafez-Whitlow [32].

8. H1 Compactness of Entropy Dissipation Measures

We now limit ourselves to the case v = 0 and the two types of domains in Figure6(a)-(b), where 1 (the solid curve in (a) and the solid closed curve in (b)) is the boundaryof the obstacle, 2 (dashed line segments in both (a) and (b)) is the far field boundary,and is the domain bounded by 1 and 2. This assumption implies = 0 on 2.In case (b), the circulation about the boundary 2 is zero.

Proposition 8.1. Let w = (u, v) be a solution to (5.1)(5.2) with u > 0 and v = 0on either domain in Figure 6(a)-(b), satisfying that q q < qcav and is bounded.Then the integral

1(

)||2 + 2() c2()

(q)2||2

dxdy

is bounded uniformly in all > 0.


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2 2

(b)(a)

1

1

Figure 6: Domains

Proof. We choose the special generator H in (7.7) which yields V of the form:

V =2

2+ P(), V = P

() =c2 q2

q2

qq

dq

q, V = . (8.1)

and (7.1) becomes

Q1(w)x + Q

2(w

)y = (1()) + q()q

dq

q(2()), (8.2)

where

Q1(w) = qH cos qH sin , Q2(w) = qH sin + qH cos .

Then

Q1(w)x + Q

2(w

)y = div

1() + 2()

qq

dq

q

+ 1()||2 2() |

|2q

dq()

d

= div

1() + 2()

qq

dq

q

+ 1()||2 + 2() c

2()

(q)2||2.

(8.3)

Integrating (8.3) over and using the divergence theorem, we obtain with q = u > 0that

(Q1(w

), Q2(w)) n ds =

1() n + 2()

qu

dq

q

nds+

1(

)||2 + 2() c2()

(q)2||2

dxdy,

:=I1 + I2,(8.4)


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where Q1, Q2 depend only on (

, ) and are independent of their derivatives. Thus,from the L bound, the left hand side of (8.4) is uniformly bounded for all > 0. Morespecifically, from (7.6) and the formula of H, we have the following:

On 1, (Q1, Q2) n = q

which is uniformly bounded in > 0;On the horizontal part of 2, = 0 and |(Q1, Q2) n| = |Q2| = |q| = 0;

On the vertical parts of 2, = 0 and |(Q1, Q2) n| = |Q1| = |(u)uH(u)| is

uniformly bounded in > 0.

Using the boundary conditions (5.2), one has

I1 =

1

2 q

u

dq

q

n ds + 2

2 u

u

dq

q

n ds=

1

|(u, v) n| qu

dq

q

ds,

which implies that I1 is uniformly bounded due to the L bound, since the integrand is

like q ln q near q = 0 and is well behaved. Therefore, (8.4) implies that I2 is uniformlybounded.

In terms of a general generator H, equation (7.1) becomes

Q1(w)x + Q2(w

)y = (1())(H H) + (2())

H +1

H

,

or

Q1(w)x + Q2(w

)y = div

1()(H H) + 2()

H +

1

H

1()

H H

2()

H +

1

H

.

(8.5)The entropy pair (Q1, Q2) satisfies (7.2) which can be written in terms of H,

Q1

= c2

qsin (H H) q cos c

2 q2q2

H +

1

H

,

Q1

= q cos (H H) q sin

H +1

H

,

Q2

=c2

qcos (H H) q sin c

2 q2q2

H +

1

H

,

Q2

= q sin (H H) + q cos

H +1

H

.

(8.6)

Proposition 8.2. Assume that q(x, y) U on U for some constant U > 0, inaddition q(x, y) q < qcavensured by the invariant regions. Then

xQ1(w) + yQ2(w) are confined in a compact set in H1(U),

for any entropy flux pair (Q1, Q2) generated by H C3. That is, hypothesis (A.2) of thecompensated compactness framework is satisfied for such entropy pair (Q1, Q2) generatedby H C3.


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Proof. For such an entropy pair (Q1, Q2) generated by H C3, equation (8.5) is satisfied.By Proposition 8.1, we have

J

1 := div

1(

)(

H H)

+ 2(

)

H +

1

H 0

in H1(U) as 0, and

J2 := 1() (H H) 2()

H +1

H

is in L1(U) uniformly in . On the other hand, (Q1(w

), Q2(w)) are uniformly bounded

which yield that J1 + J2 is bounded in W

1,(U). Then Murats lemma [31] implies thatJ1 + J

2 are confined in a compact subset of H

1(U).

As a corollary of Proposition 8.2, we conclude

Proposition 8.3. Let the viscous viscosity fields w = (u, v) have the speed q(x, y)uniformly bounded in > 0 away from zero when (x, y) is away from the obstacle boundary1, i.e. there exists a positive -independent = () 0 as 0 such that q(x, y) () for any (x, y) = {(x, y) : dist((x, y), 1) > 0}. Then

xQ1(w) + yQ2(w) are confined in a compact set in H1loc (),

for any entropy pair (Q1, Q2) generated by H C3.

9. Convergence of the Vanishing Viscosity Solutions

As noted earlier, the reduction of the support of the Young measure is accomplished

via application of the commutation identity (6.2). Here we follow a technique of Morawetz[28] and use the entropy generators obtained via classical separation variables first givenby Loewner [25]. Specifically, we look for H as the following two forms:

Hn(, ) = Fn()ein,

and

Hn(, ) = Kn()en.

Hence, from the generalized Tricomi equation (7.5) for H, we see

Fn + n2 M

2 1

Fn = 0, Kn n2 M2 1

Kn = 0,

where = dd . Substitution into the representation for Q1, Q2 as given in Proposition

7.1 generates an infinite sequence of entropy pairs (Q(n)1 , Q

(n)2 ) associated with Fn; and a

similar construction can be done for Kn.

First we apply the commutation identity to (Q(n)1 , Q

(n)2 ) associated with Fn. Set

I = (Q(n)1+ Q(n)1+

)(Q

(n)2 Q(n)2

) (Q2+ Q(n)2+

)(Q1 Q(n)1

).


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Then

I =(Fnq cos + inFnq sin )ein (Fnq cos + inFnq sin )ein

(Fnq sin + inFnq cos )ein (Fnq sin + inFnq cos )ein

(Fnq sin inFnq cos )ein (Fnq sin inFnq cos )ein

(Fnq cos inFnq sin )ein (Fnq cos inFnq sin )ein

.

With a tedious calculation, we obtain

< , I >=

, 2inq2

FnFn + FnFn

+ 2ein(

)Fnq cos Fnq sin + n2Fnq sin Fnq cos

+Fnq sin Fnq cos n2Fnq cos Fnq sin

+ 2ein(

)inFnq cos Fnq cos Fnq sin Fnq sin

Fnq sin Fnq sin Fnq cos Fnq sin

,

and then

, i

4I

=

, nq2FnFn + 12

FnFnqq sin(n( ))sin( )( + n2) nFnFnqq cos(n( ))cos( )

=

, n2

(qFn qFn)(qFn qFn)

+1

2sin2

n + 1

2( )

2nFnF

nqq

+ FnFnqq( + n2)

+1

2sin2

n 1

2( )

2nFnF

nqq

FnFnqq ( + n2)

.

Thus, from (6.2),

< , n2

(qFn qFn)(qFn qFn)

+1

2sin2

n + 1

2( )

2nFnF

nqq

+ FnFnqq( + n2)

+1

2sin2

n 1

2( )

2nFnF

nqq

FnFnqq ( + n2)

>= 0.


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Now interchange n and n to get the n2-terms equal to zero. Hence< , (qFn qFn)(qFn qFn)

+ 2

sin2n + 1

2 ( )

+ sin2n 12 ( ) FnFnqq >= 0.

(9.1)

Then we follow verbatim from the argument of Morawetz [28] to lead the followingconvergence theorem.

Theorem 9.1. Let v = 0, |u| < qcav, 1 < 3. Assume that there exists a positive-independent() 0 as 0 such that q(x, y) () for any (x, y) = {(x, y) : dist((x, y), 1) > 0}. Then the following hold:

(i) The support of the Young measure x,y strictly excludes the stagnation point q = 0and reduces to a point for a.e. (x, y) , hence the Young measure is a Diracmass;

(ii) The sequence (u, v) has a subsequence converging strongly in L2loc()2 to an en-

tropy solution in the sense of Definition 7.1;(iii) The boundary condition (u, v) n 0 on 1 is satisfied in the sense of normal

trace in [8].

Proof. First, we choose Fn qn, with n sufficiently large and q < qcr, to show that thesupport of cannot lie in both q < qcr and q > qcr as in Morawetz [27]. If the supportof lies inside q qcr, equation (9.1) or the argument of Chen-Dafermos-Slemrod-Wang[7] applies to show that is a Dirac mass. If the support of is in q qcr, Morawetzsreconstruction of DiPernas argument in [27] by using Kn associated entropies (Q1, Q2)works to show again that is a Dirac mass. The boundary condition on 1 follows theweak form of the second equation in (2.1) with R2 = (2()). The entropy inequalityfollows from (8.3).

Remark 9.1. From the boundary condition (5.2) for the viscous problem, we find that, ifone can show that 2(

) n 0 in the weak sense on 1 as 0, then we canconclude the desired boundary condition (u, v) n = 0 on 1 satisfied in the weak sense.This is the case when the limit solution has certain regularity along the boundary, i.e. anyshock strength is zero at its intersection point with the boundary, as conjectured for theshock formed in the supersonic bubble near the obstacle (cf. Morawetz [26, 29]).

Remark 9.2. For the first boundary value problem in Figure 6(a), if 1 is sufficientlysmooth, one expects that the transonic flow is strictly away from stagnation, which wouldexpect that the viscous solutions stay uniformly away from cavitation in the whole domain. However, for the second boundary value problem in Figure 6(b), one expects that thetransonic flow generally has a stagnation point on the boundary near the head of the

obstacle since the circulation about the boundary 2 is zero. The assumption that thereexists an -independent positive () 0 as 0 such that q(x, y) (), for any(x, y) , is physically motivated and designed in order to fit both cases.

10. Resolution of the Viscous Problem

As we have seen in the previous sections, it is crucial that our viscous problem possessessmooth solutions w = (u, v). We address the issue in this section. Consider the viscous


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problem: vx uy = (1()),(u)x + (v)y =

(2()

),

(10.1)

with

1 = 1, 2() =q2 c2

q2for q

2 qcr, (10.2)

and 1, 2 any smooth bounded, positive continuation for q

2qcr, and with the bound-ary conditions on the obstacle :

n = 0 on 1,2 n | (u, v) n| = 0 on 1,(u, v) (u, v) = 0 on 2, with q < qcr,

(10.3)

where is a function of q given by Bernoullis law (2.2) for > 1 and (2.5) for = 1, and

n is the unit normal pointing into the flow region on .Introduce a new variable

() :=

1

2()d.

Since () = 2() > 0, we can always invert to write = (), as well as q = q(). Theadvantage of this change of dependent variable is that we now have the viscous terms tobe and .

We use the polar velocity notation

u = q()cos , v = q()sin ,

and write the viscous system (10.1) as(q()sin )x (q()cos )y = ,(()q()cos )x + (()q()sin )y = ,

(10.4)

which yieldsq()sin x + q()cos x q()cos y + q()sin y = ,(()q()) cos x ()q()sin x + (()q()) sin y + ()q()cos y = ,

(10.5)with the boundary conditions on 1:

n = 0,

n

|()q()(cos , sin )

n

|= 0,

(10.6)

and the boundary conditions on 2:

= , = , (10.7)

where = () and are constants. It is convenient to have the homogeneousboundary conditions on 2:

:= , := ; q() := q( + ), () := ( + ).


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Hence, we have the following boundary value problem:

= q()sin( + ) x + q()cos( + ) x

q() cos( + ) y + q()sin(

+ )

y , = (()q()) cos( + ) x ()q()sin( + ) x

+(()q()) sin( + ) y + ()q()cos( + ) y,

(10.8)

with the boundary conditions on 1: n = 0, n |()q()(cos( + ), sin( + )) n| = 0,

(10.9)

and (, ) = (0, 0) on 2.Consider the classical equation

(, ) = (, ), 0

1,

which corresponds to the boundary value problem:

= q()sin( + ) x + q() cos( + ) x q()cos( + ) y + q()sin( + ) y in ,

= (()q()) cos( + ) x ()q()sin( + ) x+ (()q()) sin( + ) y + ()q()cos( + ) y in ,

n = 0 on 1, n = |()q()(cos( + ), sin( + )) n| on 1,(, ) = (0, 0) on 2,

(10.10)

which is the original system with replaced by 1.Now we recall some results from the linear elliptic theory. Consider the mixed Dirichlet-

Neumann problem

w = f1 in ,wn

= f2 on 1,

w = f3 on 2,

(10.11)

where is a bounded domain in R2 with the boundary = 12. We now introduceweighted Holder norms. For an arbitrary open set S and an arbitrary a > 0, we have theusual Holder norms a,S. If > 0 and a + b 0, we define

= {x : dist(x, 2) > } ,and

w(b)a = sup>0

a+bwa, ,

and then denote Ca(b) the set of all functions with w(b)a < , and C() = C2(1)

C().


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Lemma 10.1 (Lieberman[24], Theorem 2(b)). Suppose 1 C2+, 2 is nonempty,and satisfies the global -wedge condition and the uniform interior and exterior conecondition on . Then there is a unique solution to the linear elliptic problem (10.11) and

w()2+ Cf1(2+) + f2(1+)1+ + f3 for some > 0 and constant C > 0.

Remark 10.1. Note that Liebermans theorem is given for the boundary condition on 1:M w := iDiw + d w = f2, where d < 0. However, as noted by Lieberman on page 429 in[24], the Fredholm alternative holds, and if 2 is nonempty, the restriction d < 0 may berelaxed to d 0. The domains shown in Figure 6 satisfy the above conditions.

We define : (, ) (, ) to be the solution map associated with solving thedecoupled mixed Dirichlet-Neumann boundary value problem obtained from substitutionof (, ) into the right hand side of (10.10). For (, ) (C1+(2+))2, the associatedfunctions f1 and f2 give finite values on the right-hand side of the Schauder estimates of

Liebermans theorem and f3 = 0 in our problem. Therefore, (, ) stays in a boundedsubset of (C2+() )

2 which is a compact subset of (C1+(2+))2, and is a compact map of the

Banach space B = (C1+(2+))2 into itself.

We recall one popular version of the Leray-Schauder fixed point theorem.

Lemma 10.2 (Leray-Schauder Fixed Point Theorem). Let be a compact mapping of aBanach space B into itself, and suppose that there exists a constant M such thatWB M for all W B and all [0, 1] satisfying W = W. Then has a fixed point.

To apply the above theorem to our example, we first note that the only solution when = 0 is (, ) = (0, 0), hence it is certainly bounded in B. For = 0, we study thesolutions to our original viscous system with = 1 . The invariant region argumentwe have given provides the uniform L(

) bound independent of . The gradient estimatesfrom the entropy equality yield that (, ) are in a bounded set of (L2())2, and hence

(, ) are in the same bounded set of (L2())2 for all 0 < 1.Now we examine the equations themselves:

= f1, = f2,

where f1, f2 are in a bounded set ofL2(). Hence, for 0 < 1, (, ) are in a bounded

set ofL2() and, by the Calderon-Zygmund inequality (Theorem 9.9 in Gilbarg-Trudinger[21]), (, ) W2,2() .

Next we differentiate the both sides of our equations with respect to x, y, and we getthe equations of the form

x = f1x, y = f1y.

Since the right-hand sides are in a bounded set of L2(), so are the left-hand sides. Wecontinue in this fashion to see that there is a constant M1 (independent of m and ) suchthat

(, ) + (, )(Hm())2 M1,for all 1 m < .

Then Morreys embedding theorem applied to our case yields (, ) bounded by con-stant M1 = M in (C

s())2 for any s 1 and (, ) bounded in B for any > 0 by the


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constant M1 = M, where the constant is independent of . Thus, all possible solutions of(, ) = (, ), 0 1, are bounded by constant M1 = M with M independent of. Therefore, the hypotheses of the Leray-Schauder fixed point theorem are satisfied and

the viscous problem has a solution, i.e. we have the following theorem.Theorem 10.1. The viscous problem has a solution (u, ve) for every > 0 such thatq q < qcav for [1, 3), where q is a constant independent of > 0.

Acknowledgments. Gui-Qiang Chens research was supported in part by the NationalScience Foundation under Grants DMS-0505473, DMS-0244473, and an Alexandre vonHumboldt Foundation Fellowship. Marshall Slemrods research was supported in part bythe National Science Foundation grant DMS-0243722. Dehua Wangs research was sup-ported in part by the National Science Foundation grant DMS-0244487 and the Office ofNaval Research grant N00014-01-1-0446. The authors would like to thank the other mem-bers of the NSF-FRG on multidimensional conservation laws: S. Canic, C. D. Dafermos,

J. Hunter, T.-P. Liu, C.-W. Shu, and Y. Zheng for stimulating discussions.

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G.-Q. Chen, Department of Mathematics, Northwestern University, Evanston, IL 60208.

E-mail address: [email protected]

M. Slemrod, Department of Mathematics, University of Wisconsin, Madison, WI 53706.


D. Wang, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260.


Documents

Gui-Qiang Chen, Marshall Slemrod and Dehua Wang- Vanishing Viscosity Method for Transonic Flow