Weak solutions for a hyperbolic system with unilateral constraint and mass loss 1
F. Berthelin
2
and F. Bouchut
Universite d'Orleans, UMR 6628 Departement de Mathematiques BP 6759 45067 Orleans cedex 2, France e-mail:
[email protected] 2 Departement de Mathematiques et Applications Ecole Normale Superieure et CNRS, UMR 8553 45, rue d'Ulm 75230 Paris cedex 05, France e-mail:
[email protected] 1
Abstract
We consider isentropic gas dynamics equations with unilateral constraint on the density and mass loss. The and pressureless pressure laws are considered. We propose an entropy weak formulation of the system that incorporates the constraint and Lagrange multiplier, for which we prove weak stability and existence of solutions. The nonzero pressure model is approximated by a kinetic BGK relaxation model, while the pressureless model is approximated by a sticky-blocks dynamics with mass loss.
Key-words: conservation laws with constraint { mass loss { entropy weak
product { pressureless gas { sticky blocks Mathematics Subject Classi cation: 76T10, 35L65, 35L85, 76N15, 35A35
Solutions faibles pour un syteme hyperbolique avec contrainte unilaterale et perte de masse
Nous considerons les equations de la dynamique des gaz isentropique avec contrainte unilaterale sur la densite et perte de masse. Les lois de pression et sans pression sont considerees. Nous proposons une formulation faible entropique du systeme qui incorpore la contrainte et le multiplicateur de Lagrange, pour laquelle nous montrons la stabilite faible et l'existence de solutions. Le modele avec pression non nulle est approche par un modele de relaxation BGK cinetique, tandis que le modele sans pression est approche par une dynamique de bouchons collants avec perte de masse.
1
Contents
1 Introduction and models
2
2 Stability of the entropy weak product 3 Isentropic model with nonzero pressure
6 7
1.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 3.2 3.3 3.4 3.5 3.6
BGK model . . . . . . . . . . . . . . . . . . . . Properties of the kinetic entropy . . . . . . . . . Existence for the BGK model . . . . . . . . . . Kinetic invariant domains . . . . . . . . . . . . Relaxation limit via compensated compactness . Relaxation limit for = 3 via averaging lemma
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Sticky blocks dynamics . . . . . . . . . Properties of sticky blocks . . . . . . . Existence of a solution . . . . . . . . . Loss of the strong extremality relation
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4 Pressureless model 4.1 4.2 4.3 4.4
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7 9 11 12 13 15
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17 18 20 20
1 Introduction and models
1.1 Models
The aim of this paper is to introduce a weak formulation and establish weak stability and existence for solutions to some one-dimensional systems of conservations laws with unilateral constraint. Such system arises for example in the modeling of two-phase ows, see [8], as
(
@t + @x (u) = 0; @t (u) + @x(u + p() + ) = 0; 2
(1.1)
with constraint and pressure Lagrange multiplier 0 1; 0; (1.2) and extremality relation (1 ) = 0: (1.3) This system was studied in [20] with viscosity. Existence and weak stability of suitable weak solutions is obtained in [2] in the pressureless case p() = 0, but however, the general nonzero pressure case remains open. We refer to [22], [1], [18], [19], [16], [20] and [2] for other hyperbolic problems with constraints. Some general formulations can be found in [13]. Here we are going to consider a slightly di erent model with mass loss, that can be written as ( @t + @x(u) = Q; (1.4) @t (u) + @x (u + p()) = Qu; 2
2
with constraint and mass loss rate Lagrange multiplier 0 1; Q 0; (1.5) and extremality relation (1 )Q = 0: (1.6) This model is based on the physical idea to remove from the density what over ows with 1 as rain could make over ow a reservoir or a river. The term Qu on the right-hand side of the momentum equation expresses that the over owing matter travels at velocity u. This interpretation is especially relevant for the Saint-Venant equations when p() = . We are going to consider here pressure laws of the form p() = ; 1 < 3; 0: (1.7) The pressureless case = 0 is very particular. Existence and properties for the system of pressureless gas without constraint have been studied in [17], [9], [12], [15] and [10]. The main diculty in the model is to give a suitable sense to the products Qu in (1.4) and Q in (1.6), because Q is only a measure, and , u can be discontinuous. Indeed, since Q can be nonzero only where = 1, we have formal formulas obtained by freezing = 1 in (1.4), 2
Q = 1I @xu;
Qu = 1I (@t u + @xu );
(1.8) but of course this is again meaningless. Therefore, we provide the following entropy weak formulation of the problem, that involves what we call entropy weak products. The idea is to introduce a di erent velocity v(t; x) for the lost matter, and write the system =1
(
=1
@t + @x (u) = Q; @t (u) + @x(u + p()) = Qv; 2
with as before
2
(1.9)
0 1; Q 0: (1.10) We take v 2 L1(Q), so that the product Qv is well-de ned as a measure. We need then to formulate in a weak sense that Qv = Qu, and that Q = Q. In order to do so, we require the family of entropy weak product inequalities @t S (; u) + @x GS (; u) Q S0 (1; v) (1; v); (1.11) for any convex entropy S in a suitable family parametrized by a convex function S , where GS is its entropy ux, and S0 is its derivative with respect to (; u). Since v, by de nition, is de ned Q a.e., the term on the right-hand side of (1.11) is well-de ned. In order to see that (1.9)-(1.11) is a weak formulation of (1.4)-(1.6), we observe rst that any suitable solution to (1.4)-(1.6) also solves (1.9)-(1.11) with v = u. Conversely, if we0 have a suciently smooth solution to (1.9)-(1.11), then multiplying (1.9) by S (; u) and comparing with (1.11), we get Q S0 (; u) (1; v) Q S0 (1; v) (1; v): (1.12) 3
If we take for the entropy the physical energy
(; u) = u =2 + 1 ;
(1.13)
2
we have 0(; u) = ( =( 1) u =2; u), and (1.12) gives 1
2
#
"
Q 1 (
1) (v u) =2 0:
1
(1.14)
2
Together with the constraints (1.10), we deduce that Qv = Qu and Q = Q, except in the pressureless case = 0, in which we can only conclude Qv = Qu. We shall see in Subsection 4.4 that in this case the formulation really fails to give Q = Q in the strong sense. Our main result is that the entropy formulation of the system (1.9)-(1.11) is weakly stable. We are able to prove a priori estimates and compactness of suitable approximations, that lead to existence for the Cauchy problem. For the nonzero pressure model, the approximate solutions are obtained by a kinetic BGK equation with additional projection to enforce the constraint 1. For the pressureless model, the approximation is based on the notion of sticky blocks that has been introduced in [8] and used in [2], but here with a di erent dynamics based on mass loss. We look for solutions with regularities
2 L1t (0; 1; L1x (R) \ Lx (R));
(1.15)
1
u 2 L1t (0; 1; L1x (R)); (1.16) Q 2 M([0; 1[R); v 2 L1(Q): (1.17) The density and the momentum density u are a priori not continuous with respect to time, because Q could contain Dirac distributions in time. However, (1.8) suggests that it should not be the case, but it is an open question to decide whether or not it is the case. Thus, we consider weak solutions in the sense that for all ' 2 D([0; 1[R),
Z 1Z 0
Z 1Z 0
R
R
[@t ' + u@x'] dtdx +
Z
(x)'(0; x) dx = R 0
Z
Z
Z
[0
;1[ R
[u@t '+(u +p())@x'] dtdx+ (x)u (x)'(0; x) dx = 2
0
0
R
It includes the initial data
(0; x) = (x); (0; x)u(0; x) = (x)u (x): 0
0
0
'Q; Z
(1.18)
Z ;1[ R
[0
'Qv:
(1.19)
(1.20)
We that 2 L (R), so that we can bound a priori the mass loss, R R assume Q dtdx R dx. 0
1
0
4
1.2 Main results
The following compactness result is valid for the two possible pressure laws (isentropic model with nonzero pressure or pressureless model). Theorem 1.1 Let us consider a sequence of solutions (n; un; Qn; vn) with regularities (1.15)-(1.17) with uniform bounds in their respective spaces of (1.15)(1.17), satisfying (1.18)-(1.19) and (1.10)-(1.11). Initial data n, un are supposed to satisfy 0
0 n 1;
(n)n is bounded in L (R);
0
0
1
0
0
(1.21)
(un)n is bounded in L1(R): (1.22) In the pressureless case, we also assume that the Oleinik inequality (1.26) holds, and that Qn is bounded in L1 loc (]0; 1[; Mloc(R)). Then, up to a subsequence, as n ! 1, (n ; un; Qn; vn) *(; u; Q; v) satisfying (1.15)-(1.17), in the following sense n * ; un * u in L1 (1.23) w (]0; 1[R); Qn * Q; Qnvn * Qv in M([0; 1[R)w; (1.24) where (; u; Q; v) is a solution to (1.18)-(1.19) and (1.10)-(1.11) with initial data , u de ned by 0
0
0
0
n * in L1w(R); and nun * u in L1w(R):
(1.25) In the pressureless case, we also get (1.26). In the nonzero pressure case, we have the convergence a.e. n ! , nun ! u. We turn now to existence. The theorem is again the same for both pressure laws. Theorem 1.2 Let 2 L (R) such that 0 1 and u 2 L1(R). Then there exists (; u; Q; v) with regularities (1.15)-(1.17) satisfying (1.18)-(1.19) and (1.10)-(1.11). In the pressureless context, we have more precise results. Theorem 1.3 In addition, in the pressureless case = 0, the solution of Theorem 1.2 satis es 0
0
0
0
0
0
1
0
0
0
@x u(t; x) t ; TV a;b (u(t; :)) bt a + 2ku kL1 8a < b; essinf u (x) u(t; x) esssup u (x); Q 2 L1loc(]0; 1[; Mloc(R)); ; u 2 C (]0; 1[; L1w); @t (S (u)) + @x (uS (u)) = QS in ]0; 1[R for every S 2 C (R), where QS 2 M([0; 1[R) satis es jQS j kS kL1 jQj: 1
[
2(
]
)
0
0
0
5
(1.26) (1.27) (1.28) (1.29) (1.30) (1.31) (1.32)
The remainder of the paper is organized as follows. In Section 2, we prove the stability of the entropy weak product formulation. In Section 3, we study the non-zero pressure case. We prove the existence of solutions for a BGK model with relaxation of the type of those introduced in [6]. We obtain kinetic entropy inequalities and the existence of kinetic invariant domains, following [24], [4]. Using compensated compactness, we prove the convergence, as " ! 0, towards the nonzero pressure gas dynamics model with mass loss. Finally, we provide an alternate analysis of the BGK model in the particular case = 3 using averaging lemma. In Section 4, we study the pressureless model. We introduce the sticky blocks dynamics, and specify in this case how entropy weak product inequalities arise. Finally, we prove that the strong extremality relation is lost in weak limits for the pressureless model.
2 Stability of the entropy weak product The weak stability of the formulation (1.11) becomes clear with the two following lemmas. Lemma 2.1 Let Qn be nonpositive measures and vn 2 L1(Qn ). If (Qn)n is a sequence bounded in Mloc([0; 1[R) and (kvnkL1 Qn )n is bounded, then there exists a measure Q and a function v 2 L1 (Q) such that after extraction of a subsequence, Qn * Q, Qnvn * Qv and Qn'(vn ) * Q' Q'(v) for any convex function '. Proof. The measures Qn are bounded in Mloc, thus for a subsequence Qn * Q in Mlocw. By diagonal extraction, there exists a subsequence such that Qn'(vn) * Q' for every ' continuous. We have jQnvnj C jQnj, thus at the limit jQIdj CQ and therefore 'there exists v 2 L1 (Q) such that QId = Qv. We compare now Q and Q'(v) for ' convex. A convex function ' can be written as '(v) = supfav + b ; a; b such that ' aId + bg: Let ' be a convex function and let a; b 2 R such that ' aId + b. The measure Qn is nonpositive thus Qn'(vn) Qn(avn + b), which gives at the limit Q' Q(av'+ b). Since this is true for any a, b such that ' aId + b, we conclude that Q Q'(v). Lemma 2.2 The function v 7! S0 (1; v) (1; v) is convex for S : R ! R convex and C . Furthermore, it is a nonnegative function as soon as S 0. Proof. We have rst to specify what are the entropies S . We take the so called weak entropies, that are de ned as 0
(
)
0
1
Z S (; u) = R (; u)S ( ) d; S convex;
(2.1)
where is de ned by (3.7)-(3.9) in the case > 0, and by (; ) = ( ) if = 0 (in other words S (; u) = S (u)). Recalling that prime denotes 6
di erentiation with respect to (; u), we can express the desired quantity and get, for 1 < < 3, Z 1 0 (1 z ) S (v + a z) dz; (2.2) (1; v) (1; v) = 1
for = 3, and for = 0,
2
1
J p p S0 (1; v) (1; v) = S (v + 3) + S (v 3) =2;
(2.3)
S0 (1; v) (1; v) = S (v):
(2.4)
S
1
The result follows obviously. Remark 2.1 The sign assertion in Lemma 2.2 is helpful because the righthand side inR (1.11) becomes itself nonpositive when S 0 and we deduce the decrease of S dx.
3 Isentropic model with nonzero pressure
In this section, we introduce a kinetic BGK relaxation model that approximates the problem with nonzero pressure p() = with > 0, 1 < 3. We rst prove existence of solutions for the BGK model and establish kinetic invariant domains leading to uniform bounds. Then we let the relaxation parameter " tend to 0, and get an entropy solution to (1.9)-(1.11) via compensated compactness. For the special case = 3, an alternate proof via averaging lemma is provided.
3.1 BGK model
We consider the following kinetic BGK relaxation model, which is obtained from the one of [3], [4] by including an overall projection onto the constraint 1, (3.1) @t f + @xf = M [f"] f in ]0; 1[R R; where f = f (t; x; ) = (f (t; x; ); f (t; x; )) 2 R , t > 0, x 2 R, 2 R, f (t; x; ) 2 D ; (3.2) 0
2
1
Z
Z
(t; x) = R f (t; x; ) d; (t; x)u(t; x) = R f (t; x; ) d; M [f ](t; x; ) = M ((t; x); u(t; x); ) ; M (; u; ) = M (min(1; ); u; ); 0
1
and M is the Maxwellian de ned by
M (; u; ) = (; u); ((1 )u + ) (; u) ; (; ) = c ; a ; 2
1
2
+
7
(3.3) (3.4) (3.5) (3.6) (3.7)
J =
Z
= 2 1 ; = 1 1 12 ; c ; = a 1 1
(1 z
2
) dz
=
2 (
1)
=J;
(3.8)
p 2p = ( + 1)= ( + 3=2); a = 1 :
(3.9)
We complete (3.1) by initial data
f (0; x; ) = f (x; ); satisfying energy bounds. For 1 < < 3, we take D = D = f(f ; f ) 2 R ; f > 0 or f = f = 0g; while if = 3,
(3.10)
0
0
2
1
0
1
(3.11)
0
p
D = f(f ; f ) 2 R ; f = f and 0 f 1=2 3g; 0
2
1
1
0
0
(3.12)
and the Maxwellian simpli es in (3.13) M (; u; ) = p1 1Iu p
0
3.2 Properties of the kinetic entropy
We recall the value of the moments of M ,
Z
Z
M (; u; ) d = (; u); R
and
M (; u; ) d = (u; u + ); 2
R
Z 1 M (; u; ) d = 1 u + (; u); 2
1 R2 for every 0 and u 2 R. It results from the de nition of a convex function 2
2
0
that
Lemma 3.1 For every S : R ! R convex, we have (!
2
! )S ( ) + (! 1
1
)S (! ) + ( ! )S (! ) 2
0 if ! ! ; 0 if ! ! or ! ! : 1
2
1
2
1
9
2
2
1
(3.25)
As in [3], [4], we need a subdi erential inequality in order to prove boundedness of the entropy, namely Proposition 3.2 (Subdi erential inequality) If S : R ! R is convex, of class C , then for every 0, u; 2 R and f 2 D , we have 1
HS (f; ) HS (M (; u; ); ) + TS (; u) (f M (; u; ));
(3.26)
with
Z z ) + (a z u)S 0(u + a z ) ! 1 S ( u + a
dz; TS (; u) = J (1 z ) S 0(u + a z) (3.27) which coincides with S0 (; u) when > 0, where prime denotes di erentiation with respect to the conservative variables (; q u). We also have if f 6= 0 1
2
1
(HS0 (f; ) TS (; u)) (M (; u; ) f ) 0; (3.28) where if = 3, HS0 (f; ) = S ( )(1; 0) by convention. Proof. The case p < 3 was treated p in [4], thus let us passume that = 3. We set ! = u 3, ! = u + 3, thus ! ! = 2 3 0. We have for >0 1
2
2
p
p
1
p
p
( 3 u)S (u + p3) + ( 3 p+ u)S (u TS (; u) = p1 S (u + 3!) S (u 3) 2 3 1 ! S ( ! ) + ! S (! ) ; =! ! S (! ) S (! ) 1
2
2
2
2
1
3)
!
1
1
(3.29)
and TS (0; u) = (S (u) uS 0(u); S 0(u)). We compute
A = HS (f; ) HS (M (; u; ); ) TS (; u) (f M (; u; )) (3.30) = [f M (; u; )][S ( ) TS (; u) (1; )]: Then, if > 0, S ( ) TS (; u)(1; ) = [(! ! )S ( )+(! )S (! )+( ! )S (! )]=(! ! ); (3.31) and if = 0, S ( ) TS (; u) (1; ) = S ( ) S (u) S 0(u)( u) 0: (3.32) If ! or ! , then M (; u; ) = 0, and S ( ) TS (; u) (1; ) 0 thanks to Lemma 3.1, thus A 0. If ! < < ! , then M (; u; ) = p f and S ( ) TS (; u) (1; ) 0, thus A 0 also, which proves (3.26) and (3.28). The kinetic entropy associated to the physical energy is denoted by H = Hv = . We recall that H 0. Applying the previous result to min(1; ) and integrating in , we get a minimization principle. 0
0
2
1
2
1
1
2
2
1
0
1
2
2 2
10
0
2
1 3
2
1
0
Corollary 3.3 (Entropy minimization principle) Assume that S : R ! R is convex of class C and such that jS (v )j B (1 + v ) for some B 0. Consider f 2 L (R ) such that f 2 D a:e: and 1
2
1
Z
R
H (f ( ); ) d < 1:
(3.33)
Then S (f ( ); ) and HS (M [f ]( ); ) lie in L1 (R ), and setting (; u) = R f dHwe have R
Z
Z [f ]( ); ) d + T (min(1; ); u) (1; u)( 1) H (f ( ); ) d: H ( M S S S R R +
(3.34)
For further reference, we also provide the following obvious estimate.
Lemma 3.4 For every 0 and u; 2 R, we have 0 M (; u; ) M (; u; ):
0
0
3.3 Existence for the BGK model
We proceed as in [3] in order to apply Schauder's theorem and to get the existence of global solutions. We notice that the proofs simplify for = 3 because H (f; ) = f =2 is linear and because the kinetic system becomes in fact a rank-one model, and as a consequence we mainly only study the convergence of the rst component which is non-negative. In any case we get the following result. 0
2
Theorem 3.5 Assume that f 2 L (Rx R ) satis es f (x; ) 2 D a.e. in R R and ZZ (3.35) H (f (x; ); ) dxd = CH < 1: 0
1
0
0
0
R R
Then there exists a solution f to (3.1)-(3.12) satisfying
f 2 (C \ L1 )t([0; 1[; L (Rx R ));
(3.36)
8t 0; f (t; x; ) 2 D a:e: in R R;
(3.37)
1
8t 0;
ZZ
RR
8t 0; Z
f (t; x; ) dxd 0
ZZ RR
ZZ
RR
f (x; ) dxd; 0 0
H (f (t; x; ); ) dxd CH ; 0
Z
@t ( R f d ) + @x ( R f d ) = ( " 1) 0; 0
0
11
+
(3.38) (3.39) (3.40)
d ZZ H (f (t; x; ); ) dxd dt RR ZZ 1 H 0(f; ) ( 1 ( ) = " RR 1 ( ) + u ( 1) " 1 2 0; R where (; u) = f d and = min(1; ).
u ; u) (f M [f ]) dxd 2 2
1
2
1
(3.41)
+
R
We notice that we have a representation of the solution f , Zt 1 t=" f (t; x; ) = f (x t; )e + e s="M [f ](t s; x s; ) ds: (3.42) 0
"
0
3.4 Kinetic invariant domains
By using Corollary 3.3 and Lemma 2.2 in a computation similar to (3.41), we get Proposition 3.6 Assume that S : R ! R is convex of class C and such that 0 S (v) B (1 + v ) for some B 0. Then, with f the solution of Theorem 3.5, ZZ ZZ HS (f (t; x; ); ) dxd HS (f (x; ); ) dxd: (3.43) 1
2
0
RR
RR
Noticing that the invariant domain (3.20) is stable by the projection (; u) 7! (min(1; ); u), this allows to obtain invariant domains and bounds for , u, f and M [f ]. Theorem 3.7 For any !min < !max , the system (3.1) has the property that D~ is a family of convex kinetic invariant domains. Moreover, the set D~ is associated with the invariant domain D~ in the sense that ~ M (; u; ) 2 D~ a:e:; 8(; u) 2 D; (3.44) and for any f ( ) 2 L (R ) such that Zf ( ) 2 D~ a:e: ; (; u) 2 D~ with (; u) = f ( ) d: (3.45) R In particular, if the initial data of Theorem 3.5 satis es f (x; ) 2 D~ for a.e. x, , then, denoting by f the solution obtained in Theorem 3.5, (; u) de ned by (3.3) verify 8t 0; ((t; x); u(t; x)) 2 D~ for a.e. x. Besides f (t; x; ) 2 D~ 8t 0 and consequently f has compact support with respect to , supp f [!min ; !max ]. Furthermore , u, f , M [f ] and M [f ] are uniformly bounded in L1 . Proof. Using the functions S (v) = (v !max ) , S (v) = (!minp v) in Proposition 3.6, and the fact that 1I u < 1I!min
0
2 +
(
)2
12
3
2
2 +
3.5 Relaxation limit via compensated compactness
In this section, we prove the stability Theorem 1.1 and the existence Theorem 1.2 in the case of nonzero pressure.
Proof of Theorem 1.1 for nonzero pressure.
Let (n; un; Qn; vn) satisfy the assumptions of Theorem 1.1. Then @t S (n; un) + @x GS (n ; un) QnS0 (1; vn) (1; vn); (3.46) and since the right-hand side is bounded in Mloc, we can apply the compensated compactness result of [21] and it gives that, up to a subsequence, (n; nun) converge a.e. in ]0; 1[R when n ! 1 to some functions (; u). Using Lemmas 2.1 and 2.2, we can pass to the limit in (3.46), while the limit in (1.18)-(1.19) is obvious. We turn now to the existence result (Theorem 1.2) and prove the relaxation of (3.1) to (1.9)-(1.11). Theorem 3.8 Let us denote by f" the solution of Theorem 3.5 with the same initial data f (x; ) 2 L (R R) that satis es f (x; ) 2 D~ a.e. for some !min < !max, and the energy bound (3.35). Then (" ; u") de ned by (3.3) are uniformly bounded in L1, and passing if necessary to subsequences, ("; " u") converge a.e. in ]0; 1[R when " ! 0 to (; u), (" 1) =" * Q, (" 1) u"=" * Qv, where (; u; Q; v) have the regularities (1.15)-(1.17) R and satisfy (1.18)-(1.19) and (1.10)-(1.11), with initial data ( ; u ) = f d . Proof. The bounds of Theorem 3.7 give that ", u" f" and the support in of f" are uniformly bounded. Then, the renormalization result for a transport equation of [7] gives for any convex C function S @t HS (f"; ) + @xHS (f"; ) = HS0 (f"; ) (M [f"] f")=": By integration in , it yields 0
1
0
+
+
0
0
0
0
1
Z Z @t R HS (f"; ) d + @x R HS (f"; ) d Z 1 = (HS0 (f"; ) TS (" ; u")) (M [f"] f") d " R Z + TS (" ; u") M [f""] f" d; R with " = min(1; "). De ne Q" = (" " 1) 0: Then Z M [f"] f" d = Q" (1; u"); " R +
and
Q" TS (" ; u") = Q" TS (1; u") = Q" S0 (1; u"); 13
(3.47)
(3.48) (3.49) (3.50)
thus by (3.28)
@t ZS (" ; u") + @xGS (" ; u") @t R HS (M [f"]; ) HS (f"; ) d Z +@x HS (M [f"]; ) HS (f"; ) d R +Q"S0 (1; u") (1; u"): Next, we observe that Q" is bounded in L (]0; 1[R) since ZZ ZZ Q" dxdt f (x; ) dxd;
(3.51)
1
(3.52)
0 0
RR
as a consequence of (3.40). We deduce that (" 1) tends to 0 in L , and after extraction of a subsequence, " " ! 0 a.e. Provided that f" M [f"] ! 0 a.e. t; x; , using (3.51), we can then apply the compensated compactness result of [21] which gives that up to a subsequence, (" ; " u") (and also ("; "u")) converge a.e. in ]0; 1[R when " ! 0 to some (; u), with (; u) 2 D~ , 0 1. By applying Lemma 2.1, we get Q" * Q, Q"u" * Qv with Q 2 M([0; 1[R), v 2 L1(Q). Using again f" M [f"] ! 0 a.e. t; x; and with Lemma 2.2, the limit in (3.51) gives (1.11). A direct integration of (3.1) also gives (1.18)-(1.19) at the limit. Thus it only remains to prove that f" M [f"] ! 0 a.e. t; x; . This can be justi ed as follows. From (3.41), the integral ZZZ M [f"] f" 0 dtdxd (3.53) H (f ; ) T ( ; u ) 1
+
"
;T [RR
v2 2
]0
"
"
"
is bounded uniformly in ". Thus, if < 3, we can adapt the dissipation result of and getR that f" M [f"] ! 0 a.e. t; x; , replacing the use of the identity R ([4] f ) R " d = R (M [f" ]) d by 0
0
0
ZZZ
t;x)2B
(M [f"]) dtdxd K";
(f")
0
0
(
for any bounded set B . For = 3, the study is a little bit di erent and we refer tothe next section for the precise analysis, that leads to the same result f" M [f"] ! 0 a.e. t; x; . Proof of Theorem 1.2. Let , u satisfy 2 L (R), 0 1 and u 2 L1(R). Then there exists !min, !max such that ( ; u ) 2 D~ a.e. We take f (x; ) = M ( (x); u (x); ) 2 D~ . Since 0
0
0
1
0
0
0
0
ZZ
RR
0
0
0
Z
H (M ( ; u ; ); )dxd = R ( ; u )dx < 1; 0
0
we can apply Theorem 3.8 and we get the result.
14
0
0
3.6 Relaxation limit for
via averaging lemma
= 3
In this section, we complete the proof of Theorem 3.8 in the case = 3 by proving that f" M [f"] ! 0 a.e. t; x; also in this case, and we give an alternate compactness argument via averaging lemma instead of compensated compactness, following the ideas of [14], [23], [11], [25] and [26]. Let f" be the solution of Theorem 3.5 with the same initial data f (x; ) and ("; u") be the approximate solutions to (1.9) de ned by (3.3). In order to prove the compactness of " and "u", we use the compactness averaging lemma of [14] in the following form. Proposition 3.9 Let g" 2 L1(]0; 1[R R) satisfy 0
@t g" + @xg" = " @ ";
(3.54)
2
for some nonnegative measures ", "R locally bounded uniformly in ". If g" is boundedp in L1 uniformly in ", then R g"(t; x; ) ( ) d belongs to a compact set of Lloc (]0; 1[R), 1 < p < 1, for any 2 Cc1(R).
Proposition 3.10 The solution f" of Theorem 3.5 for = 3 satis es @t (f") + @x(f") = " @ "; 0
(3.55)
2
0
where (" )">0 and (")">0 are nonnegative measures bounded uniformly in ". Therefore, by Proposition 3.9, " and "u" are locally compact.
Proof. We set " = (M [f"] M [f"])=" R and h" = R((f") M [f"])=". We have " 0 thanks to Lemma 3.4. Since R h" d = 0, R h" d = 0 and h" has 0
0
0
0
compact support in , there exists a distribution " with compact support in such that h" = @ ". Thus we have (3.55). We integrate this equality and get ZZZ ZZ " f (x; ) dxd: 2
0 0
RR
;T [RR
]0
Take now a test function '(t; x; ) = ' (t; x)' ( ), with ' , ' nonnegative and of class Cc1, and de ne 2 C 1 by ' = @ . We have that is convex, and by the entropy minimization principle 1
2 2
2
ZZZ
h"; 'i = h@ "; ' i = 2
1
]0
;T [RR
1
' h" 0; 1
thus " 0. We integrate now (3.55) against =2, and we get 2
ZZZ
;T [RR
]0
ZZ " 2 f (x; ) dxd; 2
RR
which concludes the proof.
15
0 0
2
Proof of Theorem 3.8 when = 3.
The beginning of the proof is the same, we can replace compensated compactness by Proposition 3.10, but it remains to get that f" M [f"] ! 0 a.e. t; x; . For a subsequence, we have " ! ; "u" ! u a.e. t; x; (3.56) with (; u) 2 D~ . The bound (3.53) implies that for a subsequence, 1 h( u ) 3() i h(f ) " " " 2 Since " " ! 0 a.e., it gives 2
h
2
( u")
ih
3" (f")
2
2
0
i M (" ; u"; ) ! 0
0
a.e. t; x; : (3.57)
0
i M ("; u"; ) ! 0
a.e. t; x; ;
0
(3.58)
and we recall that this quantity is nonnegative. We set E = f(t; x) 2]0; T [R; (t; x) > 0g. On E , we have u" ! u a.e., and from (3.58), (f") ! M (; u; ) a.e. since the set of such that ( u(t; x)) = 3 (t; x) has measure zero. Then, for any bounded domain B in (t; x), by passing to the limit as " ! 0 in 2
ZZ
Z
t;x)2B\E
(
=
ZZ
t;x)2B\E
(
we get that
ZZ
t; x) 2 B t; x) 62 E
( (
0
2
Z
0
Z
ZZ
(f") dtdxd + 0
t; x) 2 B t; x) 62 E
( (
Z
M [f"] dtdxd +
ZZ
0
0
t; x) 2 B t; x) 62 E
Z
t; x) 2 B t; x) 62 E
( (
(f") dtdxd 0
Z
( (
ZZ
(f") (t; x; ) dtdxd !
M [f"] dtdxd; 0
M (; u; ) dtdxd = 0: 0
Finally we get that, up to an extraction, (f") ! M (; u; ) a.e. t; x; : (3.59) Since (f") = (f") , we also get (f") ! M (; u; ) = M (; u; ) a.e., and the result follows. 0
1
0
0
1
4 Pressureless model
0
1
This section is devoted to the proof of Theorems 1.1-1.3 when p = 0. We build a sticky blocks dynamics with mass loss that solves the system for particular data, that is used to approximate arbitrary initial data.
16
Figure 1: Collision of two blocks
t6
ul tf ul
a
t
y
H AAHH H A r HH HH A H HH H H H HH H r HH HH HH
x
ul
u
b
x
u
The analysis is similar to that of [2], and di ers from the one for the system of pressureless gases without constraint, that gives Dirac distributions on in nite time (see [5], [9], [12]). The entropies and entropy uxes are de ned by S (; u) = S (u); GS (; u) = uS (u); (4.1) for any S : R ! R convex. They satisfy S0 (1; v) (1; v) = S (v); (4.2) thus the entropy inequalities (1.11) write @t (S (u)) + @x (uS (u)) QS (v): (4.3)
4.1 Sticky blocks dynamics
Let us consider a volume fraction (t; x) and a momentum density (t; x)u(t; x) given by
(t; x) =
n X i=1
1Iai t
