Contents 1 Introduction and models 1.1 Models Main results Stability of the entropy

Transcription

1 Weak solutions for a hyperbolic system with unilateral constraint and mass loss F. Berthelin 1 and F. Bouchut 1 Universite d'orleans, UM 668 Departement de Mathematiques BP Orleans cedex, France Florent.Berthelin@labomath.univ-orleans.fr Departement de Mathematiques et Applications Ecole Normale Superieure et CNS, UM , rue d'ulm 7530 Paris cedex 05, France Francois.Bouchut@ens.fr 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 Classication: 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

2 Contents 1 Introduction and models 1.1 Models Main results Stability of the entropy weak product 6 3 Isentropic model with nonzero pressure BGK model Properties of the kinetic entropy Existence for the BGK model Kinetic invariant domains elaxation limit via compensated compactness elaxation limit for = 3 via averaging lemma Pressureless model Sticky blocks dynamics Properties of sticky blocks Existence of a solution Loss of the strong extremality relation 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 x (u) = t (u) x (u + p() + ) = 0; (1.1) with constraint and pressure Lagrange multiplier and extremality relation 0 1; 0; (1.) (1 ) = 0: (1.3) This system was studied in [0] with viscosity. Existence and weak stability of suitable weak solutions is obtained in [] in the pressureless case p() = 0, but however, the general nonzero pressure case remains open. We refer to [], [1], [18], [19], [16], [0] and [] for other hyperbolic problems with constraints. Some general formulations can be found in [13]. ere we are going to consider a slightly dierent model with mass loss, that can be written as x (u) = Q; t (u) x (u + p()) = Qu;

3 with constraint and mass loss rate Lagrange multiplier and extremality relation 0 1; Q 0; (1.5) (1 )Q = 0: (1.6) This model is based on the physical idea to remove from the density what overows with 1 as rain could make overow 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], [1], [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), Q = 1I x u; Qu = 1I =1 (@ t u x u ); (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 dierent velocity v(t; x) for the lost matter, and write the system x (u) = t (u) x (u + p()) = Qv; (1.9) with as before 0 1; Q 0: (1.10) We take v L 1 (Q), so that the product Qv is well-dened 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 t S (; u) x G S (; u) Q 0 S(1; v) (1; v); (1.11) for any convex entropy S in a suitable family parametrized by a convex function S, where G S is its entropy ux, and 0 S is its derivative with respect to (; u). Since v, by denition, is dened Q a.e., the term on the right-hand side of (1.11) is well-dened. 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 we have a suciently smooth solution to (1.9)-(1.11), then multiplying (1.9) by 0 S(; u) and comparing with (1.11), we get Q 0 S(; u) (1; v) Q 0 S(1; v) (1; v): (1.1) 3

4 If we take for the entropy the physical energy (; u) = u = + 1 ; (1.13) we have 0 (; u) = ( 1 =( 1) u =; u), and (1.1) gives " # Q 1 ( 1 1) (v u) = 0: (1.14) 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 [], but here with a dierent dynamics based on mass loss. We look for solutions with regularities L 1 t (0; 1; L 1 x () \ L 1 x()); (1.15) u L 1 t (0; 1; L 1 x ()); (1.16) Q M([0; 1[); v L 1 (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. owever, (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 ' D([0; 1[), [@ t' + u@ x '] dtdx + [u@ t'+(u +p())@ x '] dtdx+ It includes the initial data We assume that 0 Q dtdx 0 dx. 0 (x)'(0; x) dx = [0;1[ 0 (x)u 0 (x)'(0; x) dx = [0;1[ 'Q; (1.18) 'Qv: (1.19) (0; x) = 0 (x); (0; x)u(0; x) = 0 (x)u 0 (x): (1.0) L 1 (), so that we can bound a priori the mass loss, 4

5 1. 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 ; u n ; Q n ; v n ) 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 0 n, u0 n are supposed to satisfy 0 0 n 1; ( 0 n) n0 is bounded in L 1 (); (1.1) (u 0 n) n0 is bounded in L 1 (): (1.) In the pressureless case, we also assume that the Oleinik inequality (1.6) holds, and that Q n is bounded in L 1 loc(]0; 1[; M loc ()). Then, up to a subsequence, as n! 1, ( n ; u n ; Q n ; v n ) *(; u; Q; v) satisfying (1.15)-(1.17), in the following sense n * ; u n * u in L 1 w(]0; 1[); (1.3) Q n * Q; Q n v n * Qv in M([0; 1[)w; (1.4) where (; u; Q; v) is a solution to (1.18)-(1.19) and (1.10)-(1.11) with initial data 0, u 0 dened by 0 n * 0 in L 1 w(); and 0 n u0 n * 0 u 0 in L 1 w(): (1.5) In the pressureless case, we also get (1.6). In the nonzero pressure case, we have the convergence a.e. n!, n u n! u. We turn now to existence. The theorem is again the same for both pressure laws. Theorem 1. Let 0 L 1 () such that and u 0 L 1 (). 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. x u(t; x) 1 t ; (1.6) TV [a;b] (u(t; :)) (b a) + ku 0 k t L 1 8a < b; (1.7) essinf u 0 (x) u(t; x) esssup u 0 (x); (1.8) Q L 1 loc(]0; 1[; M loc ()); (1.9) ; u C(]0; 1[; L 1 w); t (S(u)) x (us(u)) = Q S in ]0; 1[ (1.31) for every S C(), where Q S M([0; 1[) satises jq S j ksk L 1jQj: (1.3) 5

6 The remainder of the paper is organized as follows. In Section, 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 [4], [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. Stability of the entropy weak product The weak stability of the formulation (1.11) becomes clear with the two following lemmas. Lemma.1 Let Q n be nonpositive measures and v n L 1 (Q n ). If (Q n ) n0 is is bounded, then a sequence bounded in M loc ([0; 1[) and (kv n k L 1 (Qn) ) n0 there exists a measure Q and a function v L 1 (Q) such that after extraction of a subsequence, Q n * Q, Q n v n * Qv and Q n '(v n ) * Q ' Q'(v) for any convex function '. Proof. The measures Q n are bounded in M loc, thus for a subsequence Q n * Q in M loc w. By diagonal extraction, there exists a subsequence such that Q n '(v n ) * Q ' for every ' continuous. We have jq n v n j CjQ n j, thus at the limit jq Id j CQ and therefore there exists v L 1 (Q) such that Q Id = 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 such that ' aid + b. The measure Q n is nonpositive thus Q n '(v n ) Q n (av n + 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. The function v 7! 0 S(1; v) (1; v) is convex for S :! convex and C 1. 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 dened as S (; u) = (; u)s() d; S convex; (.1) where is dened by (3.7)-(3.9) in the case > 0, and by (; ) = () if = 0 (in other words S (; u) = S(u)). ecalling that prime denotes 6

7 dierentiation with respect to (; u), we can express the desired quantity and get, for 1 < < 3, for = 3, and for = 0, 0 S(1; v) (1; v) = 1 J 1 The result follows obviously. 1 (1 z ) 1 S(v + a z) dz; (.) 0 S(1; v) (1; v) = S(v + p 3) + S(v p 3) =; (.3) 0 S(1; v) (1; v) = S(v): (.4) emark.1 The sign assertion in Lemma. is helpful because the righthand side in (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 t f x f = M [f] f in ]0; 1[ ; (3.1) " where f = f(t; x; ) = (f 0 (t; x; ); f 1 (t; x; )), t > 0, x,, f(t; x; ) D ; (3.) (t; x) = f 0(t; x; ) d; (t; x)u(t; x) = f 1(t; x; ) d; (3.3) M [f](t; x; ) = M ((t; x); u(t; x); ) ; (3.4) and M is the Maxwellian dened by M (; u; ) = M(min(1; ); u; ); (3.5) M(; u; ) = (; u); ((1 )u + )(; u) ; (3.6) (; ) = c ; a 1 + ; (3.7) 7

8 J = 1 1 = 1 ; = ; c ; = a =( 1) =J ; (3.8) (1 z ) dz = p ( + 1)= ( + 3=); a = p We complete (3.1) by initial data satisfying energy bounds. For 1 < < 3, we take while if = 3, 1 : (3.9) f(0; x; ) = f 0 (x; ); (3.10) D = D = f(f 0 ; f 1 ) ; f 0 > 0 or f 1 = f 0 = 0g; (3.11) D = f(f 0 ; f 1 ) ; f 1 = f 0 and 0 f 0 1= p 3g; (3.1) and the Maxwellian simplies in M(; u; ) = p 1 3 1I u p p 3<<u+ 3 K(); K() = (1; ): (3.13) In this case a single scalar equation on f 0 can be written, the second one on f 1 being proportional to it. The entropies involved in the inequality (1.11) are the so called weak entropies, dened by S (; u) = and their entropy uxes are dened by G S (; u) = (; u)s() d; S convex; (3.14) (1 )u + (; u)s() d: (3.15) To each entropy S we can associate a kinetic entropy, in the case < 3 they can be dened from a kernel by where S (f; ) = u(f; ) = f 1=f 0 1 ((f; ); u(f; ); ; v)s(v) dv; for f 6= 0; S(0; ) = 0; 0 ; (f; ) = a f 1=f 0 1! 1 + (f 0 =c ;) 1= A (3.16) (3.17) is the inverse relation for f = M(; u; ). The kernel is symmetric in ; v and satises in particular 0 and (1; v)(; u; ; v) dv = M(; u; ). For more details about this model, we refer to [4]

9 For = 3, the kinetic entropy is simply given for f = (f 0 ; f 0 ) D, by so that nally for any 1 3, S (f; ) = S()f 0 ; (3.18) S (; u) = S(M(; u; ); ) d; G S (; u) = S(M(; u; ); ) d: (3.19) Let us nally introduce the kinetic invariant domains. The macroscopic invariant domains are dened as follows for any! min <! max, ~D = f(; u) + ; = 0 or! min! 1!! max g; (3.0) where the iemann invariants! 1,! are given by! 1 = u a 1 ;! = u + a 1 : (3.1) Their corresponding kinetic invariant domains are dened in the case 1 < < 3 by where ~D = ff D; f = 0 or! min! 1 (f; )! (f; )! max g; (3.)! 1 (f; ) = u(f; ) a (f; ) 1 ;! (f; ) = u(f; ) + a (f; ) 1 ; (3.3) and in the case = 3 ~D = ff D ; 0 f 0 1 p 3 1I! min <<! max g: (3.4) 3. Properties of the kinetic entropy We recall the value of the moments of M, M(; u; ) d = (; u); and M(; u; ) d = (u; u + ); 1 M 0 (; u; ) d = 1 u + 1 (; u); for every 0 and u. It results from the denition of a convex function that Lemma 3.1 For every S :! convex, we have (!! 1 )S() + (! 1 )S(! ) + (! )S(! 1 ) 0 if! 1! ; 0 if! 1! or! 1! : (3.5) 9

10 As in [3], [4], we need a subdierential inequality in order to prove boundedness of the entropy, namely Proposition 3. (Subdierential inequality) If S :! is convex, of class C 1, then for every 0, u; and f D, we have with S (f; ) S (M(; u; ); ) + T S (; u) (f M(; u; )); (3.6) (1 z ) S(u + a z) + (a z u)s 0 (u + a z) T S (; u) = 1 1 J 1 S 0 (u + a dz; z) (3.7) which coincides with S(; 0 u) when > 0, where prime denotes dierentiation with respect to the conservative variables (; q u). We also have if f 6= 0 ( 0 S(f; ) T S (; u)) (M(; u; ) f) 0; (3.8) where if = 3, 0 S(f; ) = S()(1; 0) by convention. Proof. The case < 3 was treated in [4], thus let us assume that = 3. We set! 1 = u p 3,! = u + p 3, thus!! 1 = p 3 0. We have for > 0 T S (; u) = = 1 p p ( 3 p p u)s(u + 3) + ( 3 + u)s(u p! p 3) 3 S(u + 3) S(u p! 3) 1! 1 S(! ) +! S(! 1 ) ;!! 1 S(! ) S(! 1 ) and T S (0; u) = (S(u) us 0 (u); S 0 (u)). We compute! (3.9) A = S (f; ) S (M(; u; ); ) T S (; u) (f M(; u; )) = [f 0 M 0 (; u; )][S() T S (; u) (1; )]: (3.30) Then, if > 0, S() T S (; u)(1; ) = [(!! 1 )S()+(! 1 )S(! )+(! )S(! 1 )]=(!! 1 ); (3.31) and if = 0, S() T S (; u) (1; ) = S() S(u) S 0 (u)( u) 0: (3.3) If! 1 or!, then M 0 (; u; ) = 0, and S() T S (; u)(1; ) 0 thanks to Lemma 3.1, thus A 0. If! 1 < <!, then M 0 (; u; ) = p1 f 3 0 and S() T S (; u) (1; ) 0, thus A 0 also, which proves (3.6) and (3.8). The kinetic entropy associated to the physical energy is denoted by = v =. We recall that 0. Applying the previous result to min(1; ) and integrating in, we get a minimization principle. 10

11 Corollary 3.3 (Entropy minimization principle) Assume that S :! is convex of class C 1 and such that js(v)j B(1 + v ) for some B 0. Consider f L 1 ( ) such that f D a:e: and (f(); ) d < 1: (3.33) Then S (f(); ) and S (M [f](); ) lie in L 1 ( ), and setting (; u) = f d we have S(M [f](); ) d + T S (min(1; ); u) (1; u)( 1) + S(f(); ) d: For further reference, we also provide the following obvious estimate. (3.34) Lemma 3.4 For every 0 and u;, we have 0 M 0 (; u; ) M 0 (; u; ): 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 (f; ) = f 0 = 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. Theorem 3.5 Assume that f 0 L 1 (x ) satises f 0 (x; ) D a.e. in and (f 0 (x; ); ) dxd = C 0 < 1: (3.35) Then there exists a solution f to (3.1)-(3.1) satisfying 8t 0; f (C \ L 1 ) t ([0; 1[; L 1 (x )); (3.36) 8t 0; f(t; x; ) D a:e: in ; (3.37) f 0 (t; x; ) dxd 8t t ( f 0 d) x ( f 0 0 (x; ) dxd; (3.38) (f(t; x; ); ) dxd C 0 ; (3.39) f 0 d) = ( 1) + " 0; (3.40) 11

12 d dt = 1 " 1 " 0; (f(t; x; ); ) dxd 0 (f; ) ( 1 ( ) 1 u ; u) (f M [f]) dxd 1 ( ) 1 + u ( 1)+ (3.41) where (; u) = f d and = min(1; ). We notice that we have a representation of the solution f, f(t; x; ) = f 0 (x t; )e t=" + 1 " t 3.4 Kinetic invariant domains 0 e s=" M [f](t s; x s; ) ds: (3.4) By using Corollary 3.3 and Lemma. in a computation similar to (3.41), we get Proposition 3.6 Assume that S :! is convex of class C 1 and such that 0 S(v) B(1 + v ) for some B 0. Then, with f the solution of Theorem 3.5, S (f(t; x; ); ) dxd S (f 0 (x; ); ) dxd: (3.43) Noticing that the invariant domain (3.0) 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 and 8(; u) ~ D; M(; u; ) ~ D a:e:; (3.44) for any f() L 1 ( ) such that f() ~ D a:e: ; (; u) D ~ with (; u) = f() d: (3.45) In particular, if the initial data of Theorem 3.5 satises f 0 (x; ) ~ D for a.e. x,, then, denoting by f the solution obtained in Theorem 3.5, (; u) dened by (3.3) verify 8t 0; ((t; x); u(t; x)) ~ D for a.e. x. Besides f(t; x; ) 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 L 1. Proof. Using the functions S(v) = (v! max ) +, S(v) = (! min v) + in Proposition 3.6, and the fact that 1I ( u) <3 1I! min <<! max,]u p 3; u+ p 3[]!min ;! max [, we obtain the result by adapting the proof of [4]. 1

13 3.5 elaxation limit via compensated compactness In this section, we prove the stability Theorem 1.1 and the existence Theorem 1. in the case of nonzero pressure. Proof of Theorem 1.1 for nonzero pressure. Let ( n ; u n ; Q n ; v n ) satisfy the assumptions of Theorem 1.1. t S ( n ; u n ) x G S ( n ; u n ) Q n 0 S(1; v n ) (1; v n ); (3.46) and since the right-hand side is bounded in M loc, we can apply the compensated compactness result of [1] and it gives that, up to a subsequence, ( n ; n u n ) converge a.e. in ]0; 1[ when n! 1 to some functions (; u). Using Lemmas.1 and., 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.) 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 0 (x; ) L 1 ( ) that satises f 0 (x; ) D ~ a.e. for some! min <! max, and the energy bound (3.35). Then ( " ; u " ) dened by (3.3) are uniformly bounded in L 1, and passing if necessary to subsequences, ( " ; " u " ) converge a.e. in ]0; 1[ when "! 0 to (; u), ( " 1) + =" * Q, ( " 1) + u " =" * Qv, where (; u; Q; v) have the regularities (1.15)-(1.17) and satisfy (1.18)-(1.19) and (1.10)-(1.11), with initial data ( 0 ; 0 u 0 ) = f 0 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 1 function t S (f " ; ) x S (f " ; ) = 0 S(f " ; ) (M [f " ] f " )=": By integration in, it t S(f " ; ) d x = 1 " S(f " ; ) d (0 S(f " ; ) T S ( " ; u ")) (M [f " ] f " ) d + T S ( "; u " ) with " = min(1; " ). Dene M [f " ] f " " d; (3.47) Then and Q " = ( " 1) + " M [f " ] f " " 0: (3.48) d = Q " (1; u " ); (3.49) Q " T S ( " ; u ") = Q " T S (1; u " ) = Q " 0 S(1; u " ); (3.50) 13

14 thus by t S ( " ; u ") x G S ( " ; u ") S (M [f " ]; ) S (f " ; ) d +@ x S (M [f " ]; ) S (f " ; ) d +Q " 0 S(1; u " ) (1; u " ): (3.51) Next, we observe that Q " is bounded in L 1 (]0; 1[) since Q" dxdt f 0 0 (x; ) dxd; (3.5) as a consequence of (3.40). We deduce that ( " 1) + tends to 0 in L 1, 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 [1] which gives that up to a subsequence, ( " ; " u ") (and also ( " ; " u " )) converge a.e. in ]0; 1[ when "! 0 to some (; u), with (; u) D, ~ 0 1. By applying Lemma.1, we get Q " * Q, Q " u " * Qv with Q M([0; 1[), v L 1 (Q). Using again f " M [f " ]! 0 a.e. t; x; and with Lemma., 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 justied as follows. From (3.41), the integral 0 (f " ; ) T v ( " ; u ") M [f " ] f " dtdxd (3.53) " ]0;T [ is bounded uniformly in ". Thus, if < 3, we can adapt the dissipation result of [4] and get that f " M [f " ]! 0 a.e. t; x;, replacing the use of the identity (f " ) 0 d = (M[f " ]) 0 d by 0 (t;x)b (f" ) 0 (M [f " ]) 0 dtdxd K"; for any bounded set B. For = 3, the study is a little bit dierent and we refer to the next section for the precise analysis, that leads to the same result f " M [f " ]! 0 a.e. t; x;. Proof of Theorem 1.. Let 0, u 0 satisfy 0 L 1 (), and u 0 L 1 (). Then there exists! min,! max such that ( 0 ; u 0 ) ~ D a.e. We take f 0 (x; ) = M( 0 (x); u 0 (x); ) ~ D. Since (M( 0 ; u 0 ; ); )dxd = we can apply Theorem 3.8 and we get the result. (0 ; u 0 )dx < 1; 14

15 3.6 elaxation limit for = 3 via averaging lemma 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], [3], [11], [5] and [6]. Let f " be the solution of Theorem 3.5 with the same initial data f 0 (x; ) and ( " ; u " ) be the approximate solutions to (1.9) dened 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 " L 1 (]0; 1[ ) t g " x g " = "; (3.54) for some nonnegative measures ", " locally bounded uniformly in ". If g " is bounded in L 1 uniformly in ", then g "(t; x; ) () d belongs to a compact set of L p loc(]0; 1[), 1 < p < 1, for any C 1 c (). Proposition 3.10 The solution f " of Theorem 3.5 for = 3 t (f " ) 0 x (f " ) 0 = " ; (3.55) where ( " ) ">0 and ( " ) ">0 are nonnegative measures bounded uniformly in ". Therefore, by Proposition 3.9, " and " u " are locally compact. Proof. We set " = (M 0 [f " ] M 0 [f " ])=" and h " = ((f " ) 0 M 0 [f " ])=". We have " 0 thanks to Lemma 3.4. Since h " d = 0, h " d = 0 and h " has 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 ]0;T [ " f 0 0 (x; ) dxd: Take now a test function '(t; x; ) = ' 1 (t; x)' (), with ' 1, ' nonnegative and of class C 1 c, and dene C 1 by ' We have that is convex, and by the entropy minimization principle h " ; 'i = h@ "; ' 1 i = ]0;T [ ' 1 h " 0; thus " 0. We integrate now (3.55) against =, and we get ]0;T [ which concludes the proof. " f 0 0 (x; ) dxd; 15

16 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) ~ D. The bound (3.53) implies that for a subsequence, 1 h ( u" ) 3( ") i h (f" ) 0 M 0 ( "; u " ; ) i! 0 a.e. t; x; : (3.57) Since " "! 0 a.e., it gives h ih ( u" ) 3 " (f" ) 0 i M 0 ( " ; u " ; )! 0 a.e. t; x; ; (3.58) and we recall that this quantity is nonnegative. We set E = f(t; x) ]0; T [; (t; x) > 0g. On E, we have u "! u a.e., and from (3.58), (f " ) 0! M 0 (; 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 = (t;x)b\e (t;x)b\e (f " ) 0 dtdxd + M 0 [f " ] dtdxd + (t; x) B (t; x) 6 E (t; x) B (t; x) 6 E (f " ) 0 dtdxd M 0 [f " ] dtdxd; we get that (t; x) B (t; x) 6 E (f " ) 0 (t; x; ) dtdxd! (t; x) B (t; x) 6 E M 0 (; u; ) dtdxd = 0: Finally we get that, up to an extraction, (f " ) 0! M 0 (; u; ) a.e. t; x; : (3.59) Since (f " ) 1 = (f " ) 0, we also get (f " ) 1! M 0 (; u; ) = M 1 (; u; ) a.e., and the result follows. 4 Pressureless model This section is devoted to the proof of Theorems 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

17 Figure 1: Collision of two blocks t t f t 6 u l y AA A a u l u - A r b x x u l u r The analysis is similar to that of [], and diers from the one for the system of pressureless gases without constraint, that gives Dirac distributions on in nite time (see [5], [9], [1]). The entropies and entropy uxes are dened by S (; u) = S(u); G S (; u) = us(u); (4.1) for any S :! convex. They satisfy thus the entropy inequalities (1.11) write 4.1 Sticky blocks dynamics 0 S(1; v) (1; v) = S(v); t (S(u)) x (us(u)) QS(v): (4.3) Let us consider a volume fraction (t; x) and a momentum density (t; x)u(t; x) given by (t; x) = nx i=1 1I ai (t)<x<b i (t); (t; x)u(t; x) = nx i=1 u i (t)1i ai (t)<x<b i (t); (4.4) with a 1 (t) < b 1 (t) a (t) < b (t) a n (t) < b n (t). The time evolution is dened as follows. The number of blocks n indeed depends on t, but is piecewise constant. As long as the blocks do not meet, they move at constant velocity u i (t). When two blocks collide at a time t, the dynamics is exhibited in Figure 1, and is dened as follows. The volume fraction is given locally by (t; x) = 8 >< >: 1I al (t)<x<b l (t) + 1I ar(t)<x<br (t) if t < t ; 1I al (t)<x<b r (t) 1I a(t)<x<b(t) if t t < t f ; if t t f : (4.5) 17

18 and the momentum density u by u(t; x) = 8 >< u l 1I al (t)<x<b l (t) + u r 1I ar(t)<x<br (t) if t < t ; u l 1I al (t)<x<c(t) + u r 1I c(t)<x<br(t) if t >: t < t f ; u 1I a(t)<x<b(t) if t t f : (4.6) We have a l (t) = a + u l (t t ), b l (t) = x + u l (t t ), a r (t) = x + u r (t t ), b r (t) = b + u r (t t ), c(t) = x + u loss (t t ), a(t) = a l (t f ) + u (t t f ) and b(t) = b r (t f ) + u (t t f ), with u r u loss u l, t f = min t + (b x )=(u loss u r ); t + (x a )=(u l u loss ) ; (4.7) u = ( ul if (x a )=(u l u loss ) > (b x )=(u loss u r ); u r if (x a )=(u l u loss ) < (b x )=(u loss u r ): (4.8) One of the right or left block disappears, and in case of equality in (4.8), a(t) = b(t) and all the mass disappears. When more than two blocks collide at the same time, or if a block collides at a time t 1 with two blocks that are colliding since a time t with t < t 1 < t f, we perform the collisions locally at each interface between two blocks. We notice that with this construction, and u are both continuous in time with values in L 1 (). It only remains to dene the interface velocity u loss, and we can indeed take any relation u loss = (u l ; u r ); (4.9) where is dened for u l > u r and satises 4. Properties of sticky blocks We have the following consistency result. u r (u l ; u r ) u l : (4.10) Theorem 4.1 There exists a nonpositive measure Q(t; x) M(]0; 1[) with concentrations only in x, such that with (t; x) and u(t; x) dened by (4.4), and with the above dened dynamics, we get a solution to the weak formulation (1.18)-(1.19), (4.3), with v = u l + u r u loss. Proof. As long as there is no collision, each block moves at the constant velocity u i, and (; u) solves the pressureless Euler system. In a neighborhood of initial data, the proof is given in []. Let us now look at the case of a collision of two blocks at a time t. Let '(t; x) be a smooth function with support in of Figure 1 such that there is no other contact in. We have tf c(t) t a l (t) ( tf = S(u l ) [S(u l )@ t '(t; x) + S(u l )u x '(t; x)] dtdx " d c(t) '(t; x) dx u loss '(t; c(t)) + u l '(t; a l (t)) dt a l (t) t # dt 18

19 tf + t = S(u l ) and similarly u l '(t; c(t)) '(t; al (t)) dt ( c(tf ) a l (t f ) '(t f ; x) dx x a '(t ; x) dx + tf t (u l u loss )'(t; c(t)) dt ) ; tf br(t) t c(t) = S(u r ) [S(u r )@ t '(t; x) + S(u r )u x '(t; x)] dtdx ( br(tf ) '(t f ; x) dx b c(t f ) x '(t ; x) dx + tf t (u loss u r )'(t; c(t)) dt ) : Thus by addition tf and nally = t S(u)' (tf ; x) dx S(u)' (t ; x) dx tf h (ul u loss )S(u l ) + (u loss i u r )S(u r ) '(t; c(t)) dt; + t [S(u)@ t' + S(u)u@ x '] dtdx h@ t (S(u)) x (us(u)); 'i = t 'i x 'i = t f h (ul u loss )S(u l ) + (u loss i u r )S(u r ) '(t; c(t)) dt: Therefore we have (1.31) with t Q S = h (u l u loss )S(u l ) + (u loss u r )S(u r ) i 1I t <t<tf (x c(t)): (4.11) For S 1, we get and for S = Id, we get Q(t; x) = (u l u r )1I t <t<tf (x c(t)); (4.1) (Qv)(t; x) = (u l u r )(u l + u r u loss )1I t <t<tf (x c(t)): (4.13) Since u l > u r is necessary to have the collision, we have Q 0, and (4.13) gives v = u l + u r u loss. We prove now the entropy weak product inequality. Since QS(v) = (u l u r )S(u l + u r u loss )1I t <t<tf (x c(t)); (4.14) 19

20 we obtain, by using that u r u loss u l, which imply that u r v u l, that for S t (S(u)) x (us(u)) = Q S QS(v); (4.15) which concludes the proof. Similarly to the model with pressure Lagrange multiplier of [8], we have the discrete Oleinik entropy inequality u i (t) u i 1 (t) a i(t) b i 1 (t) t for i n: (4.16) Extending the value of u(t; x) to all x by linear interpolation between two blocks and by putting a constant at innity, we get that the sticky blocks dynamics satises Oleinik's condition (1.6). Then, we observe that according to (4.1)-(4.13), the formulas (1.8) hold, and in particular we have jqj j@ x uj; (4.17) thus Q L 1 loc(]0; 1[; M loc ()). We also have the maximum principle (1.8) and the entropy equality (1.31) with Q S (t; x) = ul u loss u l u r Thus, we get (1.3) for every S continuous. S(u l ) + u loss u r S(u u l r ) u r Q(t; x): (4.18) 4.3 Existence of a solution We do not detail the proof of Theorem 1.1, since it is very close to that of []. The same argument is used to prove that for any S continuous, n S(u n ) * S(u). Only Lemma.1 is new and gives directly the entropy weak product inequality. emark 4.1 In Theorem 1.1, we also have the existence of Q S M([0; 1[) such that (1.31) and (1.3) are satised. The proof of Theorems 1. and 1.3 is straightforward. As in [], we approximate the initial data by blocks. For this data, we build the solution by the above sticky blocks dynamics. Uniform bounds follow from the analysis of Subsection 4.. Then, we use the stability to get the solution. 4.4 Loss of the strong extremality relation This section is devoted to a counterexample that shows that in the pressureless case, the extremality relation in the strong sense (1.6) is not included in the entropy weak product inequality (1.11). We construct a sequence of sticky blocks solutions to which we can apply the stability theorem, but for which the limit does not satisfy the strong extremality relation (1 )Q = 0. We consider initially two blocks of height 1=, 0 (x) = 1 (1I 0<x<1 + 1I <x<3 ); 0 (x)u 0 (x) = 1 (1I 0<x<1 1I <x<3 ): (4.19) 0

21 We use the following approximation of these initial data, 0 n(x) = nx k=1 0 n(x)u 0 n(x) = 1I k 1 nx k=1 <x< k 1I k 1 + <x< k nx k=1 n X 1I + (k 1) k=1 1I + (k 1) <x<+ k 1 <x<+ k 1 ; (4.0) : (4.1) Using the dynamics of Section 4.1 with (u l ; u r ) = (u l + u r )=, we get a solution ( n ; u n ; Q n ; v n ) to (1.18)-(1.19), (1.10)-(1.11) with regularities (1.15)- (1.17), which is indeed given by n = n u n = nx k=1 nx k=1 1I t+ k 1 1I t+ k 1 Q n = <x<t+ k <x<t+ k nx k=1 1I x<3= + nx k=1 1I x<3= n X 1I 1 + (k 1) k=1 Now, as n! 1, one can check easily that and 1I t+ (k 1) 1I t+ (k 1) <x< t+ k 1 <x< t+ k 1 1I x>3= ; (4.) 1I x>3= ; (4.3) (x 3 <t< 1 + k 1 ); v n = 0: (4.4) Q n * 1I 1 <t< 3 (x 3 ); (4.5) n * = 1 1I t<x<t+11i x< 3 n u n * u = 1 1I t<x<t+11i x< I t<x<3 t1i x> 3 ; (4.6) 1 1I t<x<3 t1i x> 3 : (4.7) We use the stability result and we get that (; u; Q; 0) is a solution to (1.18)- (1.19), (1.10)-(1.11) with regularities (1.15)-(1.17) for the initial data 0 and 0 u 0. This solution is half of a usual sticky block solution, as can be easily checked in the pressureless case, we can multiply any weak solution by a factor between 0 and 1, it is still a solution. It can be also interpreted as the sticky block solution with constraint 1=. The two blocks with height 1= loose their mass when they collide, all the mass disappears though is staying less than 1=. The extremality relation is lost because here Q does not vanish even if < 1 everywhere. We get instead ( 1 )Q = 0. owever, for the initial data (4.19), we have not been able to nd a solution that satises the strong extremality relation, and the above weak solution could be considered as the most natural one. One could say that anyway it satises the extremality relation if we dene to take the value 1 on the line x = 3=, 1= < t < 3=. But this indicates clearly that the model introduced here is not satisfactory in the pressureless case. owever the phenomenon should not occur with pressure, as indicated by (1.14). 1

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