Non-contraction of intermediate admissible discontinuities for 3-D planar isentropic magnetohydrodynamics
aa r X i v : . [ m a t h . A P ] O c t NON-CONTRACTION OF INTERMEDIATE ADMISSIBLEDISCONTINUITIES FOR 3-D PLANAR ISENTROPICMAGNETOHYDRODYNAMICS
MOON-JIN KANG
Abstract.
We investigate non-contraction of large perturbations around intermediateentropic shock waves and contact discontinuities for the three-dimensional planar com-pressible isentropic magnetohydrodynamics (MHD). To do that, we take advantage ofcriteria developed by Kang and Vasseur in [6], and non-contraction property is measuredby pseudo distance based on relative entropy. Introduction
This article is devoted to the study of non-contraction property of certain intermediateentropic shock waves and contact discontinuities for the three-dimensional planar compress-ible inviscid isentropic MHD, which takes the form in Lagrangian coordinates: ∂ t v − ∂ x u = 0 ∂ t ( vB ) − β∂ x u = 0 ∂ t ( vB ) − β∂ x u = 0 ∂ t u + ∂ x ( p + ( B + B )) = 0 ∂ t u − β∂ x B = 0 ∂ t u − β∂ x B = 0 . (1.1)Here v denotes specific volume, and ( u , u , u ) and ( β, B , B ) represent the three-dimensionalfluid velocity and magnetic field, respectively. Those only depend on a single direction e measured by x , and no dynamics with respect to other variables. Notice that β is constantdue to the divergence-free condition of magnetic field of full MHD. As an ideal isentropicpolytropic gas, the pressure p is assumed to satisfy(1.2) p ( v ) = v − γ , γ > . The system (1.1) has a convex entropy η as η ( U ) = Z ∞ v p ( s ) ds + 12 ( u + u + u ) + 12 v ( q + q )in terms of the conservative variables U := ( v, q , q , u , u , u ) where q i := vB i for i = 2 , η is strictly convex because we consider non-vacuum states for MHD. Date : July 31, 2018.1991
Mathematics Subject Classification.
Acknowledgment.
This work was supported by the Foundation Sciences Math´ematiques de Paris as apostdoctoral fellowship, and by an AMS-Simons Travel Grant. The author thank Prof. Kevin Zumbrun forvaluable comments on stability issues for MHD.
Using the entropy η , we define its relative entropy function by η ( u | v ) = η ( u ) − η ( v ) − ∇ η ( v ) · ( u − v ) . It is well-known that the relative entropy η ( ·|· ) is positive-definite, but looses the symmetryunless η ( u ) = | u | . Nevertheless the relative entropy is comparable to the square of L distance for any bounded solutions. (See for example [6, 7, 11]) Recently in [11], Vasseur hasshown contraction for large perturbations around extremal shocks (1-shock and n -shock) ofthe hyperbolic system of conservation laws satisfying physical conditions, which is satisfiedby Euler systems of gas dynamics. In order to measure the distance between any boundedentropic solution and extremal shock, he used a spatially inhomogeneous pseudo-distanceas follows: for a given weight a >
0, the spatially inhomogeneous pseudo-distance d a isdefined by d a ( u ( t, x ) , S ( t, x )) = (cid:26) η ( u ( t, x ) | u l ) if x < σt,aη ( u ( t, x ) | u r ) if x > σt, where S ( t, x ) denotes a given extremal shock ( u l , u r , σ ), i.e., S ( t, x ) = (cid:26) u l if x < σt,u r if x > σt. Based on this pseudo-distance, it has been shown in [11] that there exists suitable weight a > α ( t ) in thesense that for all bounded entropic solution u ∈ BV loc ((0 , ∞ ) × R ) n ,(1.3) Z ∞−∞ d a ( u ( t, x + α ( t )) , S ( t, x )) dx is non-increasing in time. On the other hand for intermediate admissible discontinuities, theauthors in [6] developed criteria to identify whether the intermediate entropic shocks andcontact discontinuities are contractive or not in the pseudo-distance as above. Applying thecriteria into the two-dimensional planar isentropic MHD, it turns out in [6] that there is noweight for the contraction of certain intermediate shock waves.In this article, we use the criteria developed in [6] to show non-contraction of certain inter-mediate shocks and contact discontinuities. More precisely, we prove that there is no weight a > a .Concerning studies on stability of shock waves to the viscous model of (1.1), we refer to[1, 2, 4, 5, 10], in which it turns out that the viscous shock waves (including intermediatewaves) are Evans stable under small perturbation. This implies Lopatinski stability becauseLopatinski stability condition is necessary for stability of viscous profile (See [12]). Thussmall BV -perturbations of the inviscid MHD entropic shock waves are stable thanks toLopatinski stability (See [3, 8, 9]). Notice that this is not in contradiction with our result onnon-contraction, because our framework is based on large perturbation around the entropicdiscontinuity, thus our results says that some large perturbation can increase as time goeson. In Remark 2.1, we explain more precisely a meaning of non-contraction for entropicdiscontinuities in our framework. The rest of the paper is organized as follows. In Section 2, we present a criterion developedin [6] and six characteristic fields of the 6 × Preliminaries
Criteria on non-contraction of intermediate entropic discontinuities.
In thissection, we present a criterion in [6] for non-contraction of admissible discontinuities ofhyperbolic system of conservation laws: ∂ t u + ∂ x f ( u ) = 0 , t > , x ∈ R ,u (0 , x ) = u ( x ) . (2.4)For the system (2.4) satisfying Liu and Lax entropy conditions, the authors in [6] havedeveloped sufficient conditions to identify non-contraction of intermediate entropic discon-tinuities as follows. Theorem 2.1.
For a fixed < i < n , let ( u l , u r , σ l,r ) be a given i -th entropic discontinuitysatisfying Liu and Lax entropy conditions. Assume that there are ≤ j < i < k ≤ n suchthat j - and k -characteristic fields are genuinely nonlinear. Then the following statementsholds. • (1) For < a < , we assume that there is a C j -th rarefaction curve R ju l ( s ) with j < i such that R ju l (0) = u l and the backward curve R j, − u l ( s ) of R ju l ( s ) , i.e., λ j ( R j, − u l ( s )) < λ j ( u l ) , intersects with the ( n − -dimensional surface Σ a . Then, ( u l , u r , σ l,r ) does not satisfy contraction in the sense (1.3) with weight a . • (2) For a > , we assume that there is a C k -th rarefaction curve R ku r ( s ) with k > i such that R ku r (0) = u r and the forward curve R k, + u r ( s ) of R ku r ( s ) , i.e., λ k ( R k, + u r ( s )) >λ k ( u r ) , intersects with the ( n − -dimensional surface Σ a . Then, ( u l , u r , σ l,r ) doesnot satisfy contraction in the sense (1.3) with weight a . • (3) For a = 1 , we assume that one of the assumptions of (1) and (2) is satisfied.Then, ( u l , u r , σ l,r ) does not satisfy contraction in the sense (1.3) with weight a . Remark 2.1.
From the meaning of contraction as mentioned in (1.3) , we see the definitionof non-contraction in our framework as follows. We say that an admissible discontinuity ( u l , u r , σ ) does not satisfy contraction in the pseudo distance (1.3) with weight a if for anyLipschitz curve α ( t ) with α (0) = 0 , there are some entropy solution ¯ u and small constant T > such that for all < t < T , (2.5) Z ∞−∞ d a (¯ u ( t, x + α ( t )) , S ( t, x )) dx > Z ∞−∞ d a (¯ u (0 , x ) , S (0 , x )) dx. KANG
In fact, the authors in [6] constructed a specific (local) smooth solution ¯ u satisfying (2.5) ,which evolves from a smooth initial data ¯ u (0 , x ) = ¯ u if x ∈ ( − R, R ) for some R > ,u l if x ∈ ( −∞ , − R ) ,u r if x ∈ (2 R, ∞ ) , where ¯ u is the point appeared in Theorem 2.1, which is the intersection point of suitable rar-efaction wave and surface Σ a , thus ¯ u depends on the weight a , thus on the pseudo distance. Characteristic fields for the × system (1.1) . We here present six characteristicfields of the system (1.1). For simplicity of computation, we use non-conservative variable W := ( v, B , B , u , u , u ) and rewrite (1.1) as a quasilinear form: ∂ t W + A∂ x W = 0 , where the 6 × A is given by A := − B v − βv
00 0 0 B v − βv − c B B − β − β . where c := p − p ′ ( v ) denotes the sound speed.By a straightforward computation, we have the characteristic polynomial of A as (cid:16) λ − β v (cid:17)(cid:16) λ − (cid:16) | B | + β v + c (cid:17) λ + β v c (cid:17) = 0 , where B := ( B , B ). This equation has solutions λ = β v , α − , α + , where α ± solve thequadratic equation f (Λ) := Λ − (cid:16) | B | + β v + c (cid:17) Λ + β v c = 0 , i.e., α ± := 12 h | B | + β v + c ± r(cid:16) | B | + β v + c (cid:17) − β c v i . Then since f (Λ) = (Λ − β v )(Λ − c ) − | B | v Λ ≤ (Λ − β v )(Λ − c ) , for Λ > , (2.6)we have β v , c ∈ [ α − , α + ] . If we consider the case of | B | 6 = 0, then we have(2.7) β v , c ∈ ( α − , α + ) . Here we assume that β = 0. Thus we have six eigenvalues λ = −√ α + , λ = − β √ v , λ = −√ α − , λ = √ α − , λ = β √ v , λ = √ α + . By a straightforward computation, we have the corresponding eigenvectors r = vα + ( α + − β v ) − B − B v √ α + ( α + − β v ) − βB √ α + − βB √ α + , r = βB − βB β √ vB − β √ vB , r = vα − ( β v − α − ) B B v √ α − ( β v − α − ) βB √ α − βB √ α − ,r = − vα − ( β v − α − ) − B − B v √ α − ( β v − α − ) βB √ α − βB √ α − , r = βB − βB − β √ vB β √ vB , r = − vα + ( α + − β v ) B B v √ α + ( α + − β v ) − βB √ α + − βB √ α + , Then we can check that for each i = 1 , , ,
6, ( λ i , r i ) are genuinely nonlinear whereas ( λ , r )and ( λ , r ) are linearly degenerate as follows. Indeed since dλ · r = 12 √ α + h − ∂ v α + vα + ( α + − β v ) | {z } I + ∂ B α + B | {z } I + ∂ B α + B | {z } I i ,I = 12 h | B | + β v + p ′′ + ( | B | + β v − c )( | B | + β v − p ′′ ) + | B | c v + | B | p ′′ v r(cid:16) | B | + β v − c (cid:17) + 4 | B | c v i vα + ( α + − β v ) > | B | c v + | B | p ′′ v r(cid:16) | B | + β v − c (cid:17) + 4 | B | c v vα + ( α + − β v ) > , and for each i = 2 , I i = h v + ( | B | + β v − c ) v + c v r(cid:16) | B | + β v − c (cid:17) + 4 | B | c v i B i > c v B i r(cid:16) | B | + β v − c (cid:17) + 4 | B | c v ≥ , we have dλ · r >
0. Using the similar argument, we have dλ i · r i > i = 3 , ,
6. Onthe other hand, it is easy to get dλ i · r i = 0 for i = 2 , KANG non-contraction of shock waves In this Section, we show that there is no weight a > a .Let ( U l , U r , σ ) be any 3-shock waves satisfying the Rankine-Hugoniot condition: − [ u ] = σ [ v ] , − β [ u ] = σ [ q ] , − β [ u ] = σ [ q ] , [ p ] + h q v + q v i = σ [ u ] , − β h q v i = σ [ u ] , − β h q v i = σ [ u ] , (3.8)where [ f ] := f r − f l .By Lax condition, dλ · r > − r ( U l ) is a tangent vector at U l of the 3-shockcurve S U l issuing from U l . Thus since dv · ( − r ) < du · ( − r ) <
0, we have(3.9) [ v ] < u ] < . This is true at least for weak shock. But, this relation can be justified for shock waves ofarbitrary amplitude. We give this justification in the Appendix for the reader’s convenience.On the other hand, since it follows from (3.8) that for each i = 2 , β h q i v i = σ [ q i ], wehave B i,r = v l − β σ − v r − β σ − B i,l , equivalently, B i,r − B i,l = [ v ] β σ − − v r B i,l . Moreover, since (2.7) and Lax condition yield β v r > α − ( U r ) = λ ( U r ) > σ , and [ v ] <
0, we get B i,r − B i,l = < B i,l > > B i,l <
0= 0 if B i,l = 0 . (3.10)Notice that we do not see the sign of v l − β σ − , thus of B i,r , because we have not foundthe explicit formulation of the speed σ . Based on the observation above, we here considerthe specific condition that the 3-shock wave satisfies one of the following cases :For each i = 2 ,
3, ( B i,l , B i,r ) satisfies one of B i,l > B i,r ≥ B i,l < B i,r ≤ B i,l = B i,r = 0 . (3.11) Remark 3.1.
In the above assumption (3.11) , when B ,l = 0 and B ,l = 0 simultaneously,it follows from (2.6) that { β v l , − p ′ ( v l ) } = { α − ( U l ) , α + ( U l ) } . If β v l > − p ′ ( v l ) , i.e., α + ( U l ) = β v l , then λ ( U l ) = λ ( U l ) = − β √ v l But, the eigenspacecorresponding to the eigenvalue − β √ v l is spanned by independent two eigenvectors: (0 , , , , √ v l , √ v l ) T , (0 , , − , , √ v l , −√ v l ) T . This implies that dλ ( U ) · r ( U ) = 0 except for U = U l as an umbilical point. Likewise, if β v l < − p ′ ( v l ) , i.e., α − ( U l ) = β v l , then dλ ( U ) · r ( U ) = 0 except for U = U l . Thus for thosesingular cases, the 1- and 3-characteristic fields are still genuinely nonlinear. Similarly forthe case of B ,r = 0 and B ,r = 0 , the 4- and 6-characteristic fields are genuinely nonlinearas well. For a given 4-shock wave ( ˜ U l , ˜ U r , σ ), using the same arguments as above, we have [˜ v ] > u ] <
0, and ˜ B i,l − ˜ B i,r = [˜ v ]˜ v l − β σ − ˜ B i,l . Since (2.7) and Lax condition yield β v l > α − ( U l ) = λ ( U l ) > σ , we have ˜ B i,l − ˜ B i,r = < B i,r > > B i,r <
0= 0 if B i,r = 0 . Thus we consider the analogous condition that the 4-shock wave satisfies one of the followingcases : For each i = 2 ,
3, ( ˜ B i,l , ˜ B i,r ) satisfies one of˜ B i,r > ˜ B i,l ≥ B i,r < ˜ B i,l ≤ B i,l = ˜ B i,r = 0 . (3.12)We are now ready to show that for any a >
0, there is no a -contraction of such interme-diate shocks as follows. Theorem 3.1.
Let ( U l , U r , σ ) be a given 3-shock wave of the system (1.1) - (1.2) satisfying (3.11) . Then there is no weight a > such that ( u l , u r ) satisfies contraction in the sense (1.3) with weight a . Likewise, this result holds for a given 4-shock wave ( ˜ U l , ˜ U r , σ ) satisfying (3.12) .Proof. • Case of 3-shock wave:
First of all, we show that for any 0 < a <
1, the backward1-rarefaction wave R , − U l issuing from U l intersects with the hypersurface Σ a (with dimension5), i.e., Σ a := { U | η ( U | U l ) = aη ( U | U r ) } . KANG
Since dv · r = vα + ( α + − β v ) > v is strictly monotone along the integral curve of the vectorfield r , which means that the 1-rarefaction wave can be parameterized by v . Moreoversince dλ · r > − r is the tangent vector field of the backward 1-rarefaction wave R , − U l ,which implies that v decreases along R , − U l . That is, v + ≤ v l for all parameters v + of R , − U l . Notice that R , − U l is well-defined for all v + ∈ (0 , v l ], because − r ( W ) is smooth for all W ∈ (0 , ∞ ) × R .In order to claim that R , − U l intersects with Σ a for any a <
1, we use the fact thatfor a < , η ( U | U l ) ≤ aη ( U | U r ) is equivalent to η ( U ) ≤ − a ( η ( U l ) − aη ( U r ) − ∇ η ( U l ) · U l + a ∇ η ( U r ) · U r + ( ∇ η ( U l ) − a ∇ η ( U r )) · U ) , (3.13)which is rewritten as(3.14) Z ∞ v p ( s ) ds + 12 ( u + u + u ) + 12 v ( q + q ) ≤ c + c ( v + q + q + u + u + u ) , for some constants c , c . This implies that η ( U | U l ) ≤ aη ( U | U r ) ⇐⇒ v > c ∗ and | q | + | q | + | u | + | u | + | u | ≤ c ∗ for some constants c ∗ , c ∗ > , since R ∞ p ( s ) ds = + ∞ , and all positive terms on u i and q i are quadratic in the left-handside, whereas linear in the right-hand side of (3.14). Therefore there exists 0 < v ∗ ≪ c ∗ such that η ( R , − U l ( v ∗ ) | U l ) > aη ( R , − U l ( v ∗ ) | U r ) , which implies that R , − U l intersects with Σ a for a <
1, because R , − U l is a continuous curveissuing from U l ∈ { U | η ( U | U l ) < aη ( U | U r ) } .On the other hand, we claim that the forward 6-rarefaction wave R , + U r issuing from U r intersects with the surface Σ a for any a ≥ dλ · r > dv · r = − vα + ( α + − β v ) < r is the tangent vector of the forward 6-rarefaction wave R , + U r , and the parameter v + decreases along R , + U r . Moreover R , + U r is well-defined for all v + ∈ (0 , v r ].For any fixed a ≥
1, we consider a continuous functional F a defined by F a ( U ) := η ( U | U l ) − aη ( U | U r ) . We first show that the functional F (when a = 1) satisfies(3.16) F ( R , + U r ( v ∗ )) < v ∗ ∈ (0 , v r ] . Since ∇ η ( U ) = (cid:16) − p − q + q v , q v , q v , u , u , u (cid:17) T , we use (3.15) to compute dF ( R , + U r ( v + )) dv + = ( ∇ η ( U r ) − ∇ η ( U l )) · dR , + U r ( v + ) dv + = (cid:16) [ p ] + h q + q v i(cid:17) v + α + ( α + − βv + ) + [ u ] v + √ α + ( α + − βv + )+ X i =2 (cid:16)h q i v i q i + v + − [ u i ] βq i + v + √ α + (cid:17) . (3.17)Using (3.8), we have dF ( R , + U r ( v + )) dv + = [ u ]( σ + √ α + ) v + α + ( α + − βv + ) + X i =2 h q i v i q i + v + (cid:16) β σ √ α + (cid:17) = [ u ] (cid:16) v + √ α + − β √ α + + σ ( v + − βα + ) (cid:17)| {z } I + X i =2 h q i v i q i + v + (cid:16) β σ √ α + (cid:17)| {z } I . Since (1.2) and (2.7) yields(3.18) v + √ α + > v + p − p ′ ( v + ) = q γv − γ +1+ → ∞ as v + → , it follows from (3.9) that I → −∞ as v + → I , we consider one of conditions in (3.11). If B ,l > B ,r ≥
0, i.e., h q v i <
0, wehave q ≥ R , + U r ( v + ) because of q ,r = v r B ,r ≥ dB · r = B = > B > < B <
0= 0 if B = 0 . Moreover, since α + → + ∞ as v + →
0+ by (3.18), we have h q v i q v + (cid:16) β σ √ α + (cid:17) ≤ v + ≪ . This is also true in the case of B ,l < B ,r ≤
0, i.e., h q v i >
0, because of q ≤ h q v i q v + (cid:16) β σ √ α + (cid:17) ≤ v + ≪ . Thus the condition (3.11) yields I ≤ v + ≪ , which yields dF ( R , + U r ( v + )) dv + → −∞ as v + → , which implies (3.16).Therefore we conclude that R , + U r intersects with Σ a for any a ≥
1, because F a ( U r ) > F a ( R , + U r ( v ∗ )) < F ( R , + U r ( v ∗ )) < a ≥ a >
0, the 3-shock wave ( U l , U r , σ ) does not satisfies contraction in thesense (1.3) with weight a thanks to Theorem 2.1. • Case of 4-shock wave :
Following the same arguments as above in a symmetric way,we have the non-contraction for 4-shock wave ( ˜ U l , ˜ U r , σ ) satisfying (3.12). More precisely,we can show that the backward 1-rarefaction wave R , − ˜ U l intersects with˜Σ a := { U | η ( U | ˜ U l ) = aη ( U | ˜ U r ) } for any 0 < a ≤ , and the forward 6-rarefaction wave R , +˜ U r intersects with ˜Σ a for any a >
1. We omit thedetails. (cid:3)
Remark 3.2.
In the proof of Theorem 3.1, we used the condition (3.11) only to ensure theintersection of R , + U r with the hyperplane Σ , and similarly the condition (3.12) only for theintersection of R , − ˜ U l with ˜Σ . In other words, R , − U l (resp. R , − ˜ U l ) intersects with Σ a (resp. ˜Σ a ) for a < , and R , + U r (resp. R , +˜ U r ) intersects with Σ a (resp. ˜Σ a ) for a > without thecondition (3.11) (resp. (3.12) ). non-contraction of contact discontinuities We here show that there is no weight a > a .Let ( U l , U r ) be a given 2-contact discontinuity (or 5-contact discontinuity) of the system(1.1)-(1.2). Since i -contact discontinuity ( U l , U r ) is a integral curve of the vector field r i foreach i = 2 ,
5, we have(4.19) [ v ] = 0 and [ u ] = 0 , which implies that [ B i ] = 0 for some i = 2 ,
3, otherwise U l = U r .Contrary to the case of shock waves, there is no sign of [ B ] and [ B ] because of σ ≡ β v l = β v r , which is due to the degeneracy of contact discontinuity.We here present non-contraction of contact discontinuities satisfying either ( A ) or ( B ):( A ) : For each i = 2 ,
3, ( B i,l , B i,r ) satisfies one of B i,r > B i,l > B i,r < B i,l < , ( B ) : For each i = 2 ,
3, ( B i,l , B i,r ) satisfies one of B i,l > B i,r > B i,l < B i,r < . (4.20) Theorem 4.1.
Let ( U l , U r ) be a given 2-contact discontinuity (or 5-contact discontinuity) ofthe system (1.1) - (1.2) . Then there is no weight a > such that ( u l , u r ) satisfies contractionin the sense (1.3) with weight a .Proof. We follow the same arguments as the proof of Theorem 3.1. First of all, we can seethat the backward 1-rarefaction wave R , − U l issuing from U l intersects with Σ a for a < R , + U r issuing from U r intersects with Σ a for a > R , − U l and R , + U r intersects with the Σ , we consider a functional F ( U ) := η ( U | U l ) − η ( U | U r ) . • Case (A) (For each i = 2 ,
3, ( B i,l , B i,r ) satisfies one of B i,r > B i,l > B i,r < B i,l < R , − U l parametrized as v + intersects with Σ . First ofall, using the same computation as (3.17) with(4.19), we have dF ( R , − U l ( v + )) dv + = X i =2 h q i v i q i + v + (cid:16) − β σ √ α + (cid:17) . If B ,r > B ,l >
0, i.e., h q v i >
0, since − r is the tangent vector field of the backward1-rarefaction wave R , − U l , and dB · ( − r ) = B , we have q v + → ∞ as v + → h q v i q v + (cid:16) β σ √ α + (cid:17) → ∞ for v + → , where we have used the fact that α + → ∞ as v + →
0+ by (3.18).This is also true in the case of B ,r < B ,l <
0, because of h q v i < q v + → −∞ as v + → h q v i q v + (cid:16) β σ √ α + (cid:17) → ∞ for v + → . Thus we have dF ( R , − U l ( v + )) dv + → ∞ as v + → , which implies that R , − U l intersects with Σ , because F ( U l ) < F ( R , − U l ( v ∗ )) > v ∗ ≪ • Case (B) (For each i = 2 ,
3, ( B i,l , B i,r ) satisfies one of B i,l > B i,r > B i,l < B i,r < R , + U r intersects with Σ under thoseconstraints. (cid:3) appendix We here present that the relations (3.9) holds true for 3-shock waves of arbitrary ampli-tude. Using (3.8) and the entropy inequality with the entropy flux G := (cid:16) p + q + q v (cid:17) u − βv ( q u + q u ) , we have0 ≥ [ G ] − σ [ η ]= [ u ] p l + u ,r [ p ] + X i =2 (cid:16)
12 [ u ] q i,l v l + u ,r h q i v i(cid:17) − σ [ u ] u ,r + u ,l − X i =2 (cid:16) β h q i u i v i + σ u i ]( u i,r + u i,l ) (cid:17) − σ X i =2 h q i v i − σ Z v l v r p ( s ) ds = [ u ] p l + 12 [ u ] X i =2 q i,l v l + u ,r [ p ] + u ,r X i =2 h q i v i − (cid:16) [ p ] + X i =2 h q i v i(cid:17) u ,r + u ,l − β X i =2 (cid:16) [ u i ] q i,l v l + h q i v i u i,r (cid:17) + β X i =2 h q i v i ( u i,r + u i,l )+ σ X i =2 (cid:16) q i,l v r v l [ v ] − [ q ] v r ( q i,r + q i,l ) (cid:17) − σ Z v l v r p ( s ) ds = [ u ] p l + 12 [ u ] X i =2 q i,l v l + [ u ]2 [ p ] + [ u ]2 X i =2 h q i v i − X i =2 β [ u i ] (cid:16) q i,l v l + 12 h q i v i(cid:17) + σ X i =2 (cid:16) q i,l v r v l [ v ] − [ q i ] v r ( q i,r + q i,l ) (cid:17) − σ Z v l v r p ( s ) ds, using (3.8) again,= − σ [ v ] p l + σ v ] X i =2 q i,l v l − σ v ][ p ] + σ v ] X i =2 h q i v i + X i =2 σ [ q i ] (cid:16) q i,l v l + 12 h q i v i(cid:17) + σ X i =2 (cid:16) q i,l v r v l [ v ] − [ q i ] v r ( q i,r + q i,l ) (cid:17) − σ Z v l v r p ( s ) ds = − σ (cid:16)(cid:16) [ v ]2 ( p r + p l ) − Z v r v l p ( s ) ds (cid:17) + X i =2 (cid:16) [ v ]2 (cid:16) q i,r v r + q i,l v l − q i,l v r v l (cid:17) − [ q i ]2 (cid:16) q i,l v l + q i,r v r − q i,r + q i,l v r (cid:17)(cid:17)(cid:17) = − σ [ v ] (cid:16) Z (cid:16) sp r + (1 − s ) p l − p ( sv r + (1 − s ) v l ) (cid:17) ds + X i =2 (cid:16) q i,r v r − q i,l v l (cid:17) (cid:17) . Finally, using the convexity of p , we have σ [ v ] ≥ . Thus by Lax condition σ < λ ( U l ) <
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Laboratoire Jacques-Louis Lions,University Pierre et Marie Curie, Paris, France
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