GitBucket
Newer
zeynalig
<?xml version="1.0" ?> <tei> <teiHeader> <fileDesc xml:id="-1"/> </teiHeader> <text xml:lang="en"> <front>A SIMPLE DERIVATION OF BV BOUNDS FOR INHOMOGENEOUS <lb/>RELAXATION SYSTEMS * <lb/> BENO IT PERTHAME † , NICOLAS SEGUIN ‡ , AND MAGALI TOURNUS § <lb/> Abstract. We consider relaxation systems of transport equations with heterogeneous source <lb/>terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail <lb/>in this context and in particular BV estimates. They are the most standard and simplest way to <lb/>prove compactness and convergence. <lb/>We provide a novel and simple method to obtain partial BV regularity and strong compactness <lb/>in this framework. The standard notion of entropy is not convenient either and we also indicate <lb/>another, but closely related, notion. We give two examples motivated by renal flows which consist <lb/>of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general. <lb/> Key words. Hyperbolic relaxation; spatial heterogeneity; entropy condition; boundary condi-<lb/>tions; strong compactness. <lb/> Subject classifications. 35L03, 35L60, 35B40, 35Q92 <lb/></front> <body> 1. Introduction <lb/> The usual framework of hyperbolic relaxation [4, 13, 3] concerns the convergence <lb/>of a general hyperbolic system with stiff source terms toward a conservation law, <lb/>when the relaxation parameter goes to zero. More specifically, Jin and Xin [10] <lb/>introduced a 2 × 2 linear hyperbolic system with stiff source term that approximates <lb/>any given conservation law. The problem of interest is to prove the convergence of the <lb/>microscopic quantities depending on toward the macroscopic quantities. A problem <lb/>entering the framework of hyperbolic relaxation is motivated by very simplified models <lb/>of kidney physiology [18, 19, 17] and fits the Jin and Xin framework with two major <lb/>differences, boundary conditions, and spatial dependence of the source term. The <lb/>type of boundary condition and the spatial dependence constitute the main novelty <lb/>of the present study. <lb/>The system represents two solute concentrations u (x,t) and v (x,t) and is writ-<lb/>ten, for t ≥ 0 and x ∈ [0,L], <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> ∂u <lb/> ∂t <lb/> (x,t)+ <lb/> ∂u <lb/> ∂x <lb/> (x,t) = <lb/>1 <lb/> <lb/> <lb/> h(v (x,t),x)) − u (x,t) <lb/> <lb/> , <lb/> ∂v <lb/> ∂t <lb/> (x,t) − <lb/> ∂v <lb/> ∂x <lb/> (x,t) = <lb/>1 <lb/> <lb/> <lb/> u (x,t) − h(v (x,t),x)) <lb/> <lb/> , <lb/>u (0,t) = u 0 , <lb/>v (L,t) = αu (L,t), <lb/> α∈ (0,1), <lb/> u (x,0) = u 0 (x) > 0, <lb/> v (x,0) = v 0 (x) > 0. <lb/>(1.1) <lb/> </body> <front>* Received date / Revised version date <lb/> † Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, <lb/>F-75005, Paris, <lb/>CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, <lb/>INRIA Paris-Roquencourt, EPC Mamba (benoit.perthame@upmc.fr). <lb/> ‡ Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, <lb/>F-75005, Paris, <lb/>CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, (nicolas.seguin@upmc.fr). <lb/> § Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, <lb/>F-75005, Paris <lb/>CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France <lb/>Present address : Department of Mathematics, Pennsylvania State University, University Park, <lb/>Pennsylvania 16802, USA, (magali.tournus@ann.jussieu.fr). <lb/></front> <page>1 <lb/></page> <page>2 <lb/></page> <note place="headnote">A simple detour for BV bounds in relaxations systems <lb/></note> <body>The question of interest here is to understand the behavior of u and v when <lb/> vanishes. Another question, addressed in [16], is to explain the urine concentration <lb/>mechanism due to the 'pumps', on the cell membranes, represented by the nonlinearity <lb/> h(v,x). <lb/> We make the following hypotheses <lb/> h(0,x) = 0, <lb/>1 < β ≤ <lb/> ∂h <lb/>∂v <lb/> (v,x) ≤ µ, <lb/> (1.2) <lb/>sup <lb/> v <lb/> L <lb/> 0 <lb/> | <lb/> ∂h <lb/>∂x <lb/> (v,x)|dx ≤ C, <lb/>h(.,x) is not locally affine. <lb/>(1.3) <lb/>Here C, β and µ are positive constants. Also, we use the following bounds on the <lb/>initial data <lb/> u 0 , v 0 ∈ L ∞ (0,L), <lb/> d <lb/>dx <lb/>u 0 ∈ L 1 (0,L), <lb/> d <lb/>dx <lb/>v 0 ∈ L 1 (0,L), <lb/>(1.4) <lb/>and, the well-prepared initial data condition <lb/> u 0 (x) = h(v 0 (x),x), <lb/> x∈ [0,L]. <lb/>(1.5) <lb/>It is simple and standard to find the formal limit as vanishes. Adding the two <lb/>equations of (1.1), we find the conservation law <lb/> ∂[u + v ](x,t) <lb/> ∂t <lb/> + <lb/> ∂[u − v ](x,t) <lb/> ∂x <lb/> = 0. <lb/>(1.6) <lb/>We expect that, in the limit, the idendity holds <lb/> u(x,t) = h(v(x,t),x) <lb/> and this identifies the limiting quasi-linear conservation law <lb/> ∂[h(v(x,t),x)+ v(x,t)] <lb/> ∂t <lb/> + <lb/> ∂[h(v(x,t),x) − v(x,t)] <lb/> ∂x <lb/> = 0, <lb/>(1.7) <lb/>and the boundary conditions on ρ(x,t) = h(v(x,t),x)+ v(x,t), namely ρ(0,t) = u 0 + <lb/> h −1 (u 0 ,0) and ρ(L,t) is free, can be identified via the standard argument of Bardos-<lb/>Le Roux and Nédélec, [2], using the adapted Kružkov entropies that we introduce in <lb/>Section 3. <lb/>The justification poses particular difficulties due to the boundary condition and <lb/> x-dependent flux. <lb/>One difficulty is that, in order to justify the above limit, an entropy identity is <lb/>needed for two reasons. Firstly to prove that equilibrium is reached [20]. Secondly, <lb/>because at the limit = 0, a quasi-linear equation arises, shocks can be produced. <lb/>Therefore, an entropy formulation is needed to define uniquely the solutions. When <lb/>applied to this conservation equation, usual convex entropies as both in [5] or Kru˜ kov <lb/>[11], contain a term with a derivative of h with respect to x. When the regularity of h <lb/> is limited to BV, h x is a measure and the weak entropy formulation is not well defined. <lb/>One of our goal is to present a convenient notion of entropy with x-dependent relax-<lb/>ation in order to get rid of this problem. We use an entropy formulation introduced <lb/>in [1] and adapted to scalar conservation laws with spatial heterogeneities. <lb/> <note place="headnote"> Benoˆ t Perthame, Nicolas Seguin, Magali Tournus <lb/></note> <page> 3 <lb/></page> Another difficulty is that, in order to prove the validity of the limit, strong com-<lb/>pactness is also needed. Several methods have been developed in this goal. Uniform <lb/>bounds in the space of functions with bounded variation (BV in short) is the strongest <lb/>method and the most standard. Such bounds are proved in [14] for a similar system <lb/>with boundary conditions in the homogeneous case; the result is extended with a <lb/>source term in [21]. More elaborate tools are the compensated compactness method <lb/>[12] or the kinetic formulation and averaging lemmas [8, 15] (in particular extending <lb/>the measure valued solutions of DiPerna [7]). These methods are weaker, because <lb/>they give only convergence but no bounds, and thus apply to more general situations <lb/>than BV bounds. Because of the spatial x-dependence of h, the BV framework is not <lb/>available as today for (1.1). Whereas time BV estimates follow immediately from the <lb/>equations, compactness in the spatial direction cannot be obtained in this way. We <lb/>propose a new method to prove spatial compactness. It does not use BV bounds on <lb/>each component, but gives BV bounds in x for a single quantity and can be applied <lb/>to spatially heterogeneous systems. We are going to prove that <lb/> Proposition 1.1. We make assumptions (1.2), (1.3), (1.4), (1.5) and fix a time T . <lb/>We have <lb/>(i) u and v are uniformly bounded in L ∞ ([0,L] × [0,T ]), <lb/> (ii) ∂ <lb/>∂t u , ∂ <lb/>∂t v and ∂ <lb/>∂x [u − v ] are bounded in L ∞ ([0,T ];L 1 (0,L)), <lb/> (iii) there exists v ∈ L ∞ ([0,L] × [0,T ]) such that <lb/> u (x,t) −→ <lb/> →0 <lb/> h(v(x,t),x), <lb/>v (x,t) −→ <lb/> →0 <lb/> v(x,t), a.e. <lb/> (1.8) <lb/> (iv) the equation (1.7) is satisfied and entropy inequalities hold, see section 3. <lb/> However it is not correct that ∂ <lb/>∂x u and ∂ <lb/>∂x v are separately bounded in <lb/> L ∞ ([0,T ];L 1 (0,L)). <lb/> 2. L ∞ bound <lb/> We first prove uniform estimates. Unlike the homogeneous case, L ∞ bounds are <lb/>not always true and the most general existence theory is in L 1 , see [6]. Here we build <lb/>particular sub and supersolutions of (1.1) which are uniformly bounded in . <lb/> Lemma 2.1. The solution of (1.1) satisfies the uniform estimate <lb/> u L ∞ ([0,L]×[0,T ]) ≤ K(β,u 0 ,u 0 ,v 0 ), <lb/> v L ∞ ([0,L]×[0,T ]) ≤ K(β,u 0 ,u 0 ,v 0 ). (2.1) <lb/> Proof. To obtain an L ∞ bound on the time-dependent solution, we follow the <lb/>approach of [1] and use the comparison principle with appropriate supersolution. <lb/>Indeed, because of the x−dependence of h, constant functions are not super-solution <lb/>of the stationary problem. We introduce the stationary version of (1.1) <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> dU <lb/> dx <lb/> (x) = <lb/>1 <lb/> <lb/> <lb/> h(V (x),x)) − U (x) <lb/> <lb/> , <lb/> − <lb/> dV <lb/> dx <lb/> (x) = <lb/>1 <lb/> <lb/> <lb/> U (x) − h(V (x),x)) <lb/> <lb/> , <lb/>U (0) = U 0 > 0, <lb/> V (L) = αU (L). <lb/>(2.2) <lb/>We are going to prove that there exists a smooth super solution (U ,V ) of the <lb/>stationary problem (2.2), and a constant K(U 0 ,β) > 0 such that <lb/>0 ≤ U (x) ≤ K(U 0 ,β), <lb/> 0 ≤ V ≤ K(U 0 ,β). <lb/> (2.3) <lb/> <page> 4 <lb/></page> <note place="headnote">A simple detour for BV bounds in relaxations systems <lb/></note> This concludes the proof of Lemma 2.1 because a solution of (2.2) where U 0 ≥ u 0 , is a <lb/>super-solution of (1.1), and the comparison principle gives 0 ≤ u ≤ U and 0 ≤ v ≤ V . <lb/>Because it has been proved in [18] that (2.2) admits a unique solution which lies <lb/>in BV([0,L] × [0,T ]) (fixed point argument for the existence and contraction for the <lb/>uniqueness), it remains to prove that a solution of (2.2) with U 0 ≥ u 0 is uniformly <lb/>bounded in . <lb/> Adding the two lines of (2.2), we obtain a quantity which does not depend on x, <lb/>U (x) − V (x) =: K . <lb/> (2.4) <lb/>Using the boundary values, we find uniform bounds on K <lb/> K = U 0 − V (0) ≤ U 0 <lb/> K = U (L) − V (L) = (1− α)U (L) ≥ 0, <lb/>(2.5) <lb/>and thus <lb/>0 ≤ K ≤ U 0 , <lb/>U (L) ≤ <lb/> U 0 <lb/> 1 − α <lb/> . <lb/> (2.6) <lb/>Hence, we just have to prove that U is uniformly bounded in L ∞ , knowing that U 0 <lb/> and U (L) are uniformly bounded in R. For that, we use the maximum principle <lb/>assuming C 1 regularity (one can easily justify it for a regularized function h(v,x) and <lb/>pass to the limit). Indeed, if U reaches its maximal value on the boundary, the result <lb/>follows from (2.6). If U reaches its maximal value at x 0 ∈]0,L[, then, <lb/>0 = <lb/> dU <lb/> dx <lb/> (x 0 ) = <lb/>1 <lb/> <lb/> <lb/> h(U (x 0 ) − K ,x 0 ) − U (x 0 ) <lb/> <lb/> , <lb/> (2.7) <lb/>and thus by assumption (1.2), <lb/> U (x 0 ) = h(U (x 0 ) − K ,x 0 ) ≥ βU (x 0 ) − βK . <lb/> (2.8) <lb/>Finally, (2.3) is proved because the above inequality gives <lb/> U (x 0 ) ≤ <lb/> β <lb/>β − 1 <lb/> K . <lb/> (2.9) <lb/> 3. Adapted (heterogeneous) entropies <lb/> As explained in the introduction, entropies are useful to derive additional bounds <lb/>and to characterize the limit as vanishes, see [9, 4, 20]. We are going to use specific <lb/>entropies adapted to spatial dependence. <lb/>We recall the usual approach which is to define <lb/> Definition 3.1 (Entropy pair). We call an entropy pair for the system (1.1) a <lb/>couple of functions (S, Σ) in BV <lb/> <lb/> [0,1],C(R) <lb/> <lb/> which satisfy <lb/> (i) S(.,x) and Σ(.,x) are convex, <lb/> (ii) <lb/> ∂S <lb/>∂u <lb/> (h(v,x),x) = <lb/> ∂Σ <lb/>∂v <lb/> (v,x). <lb/>(3.1) <lb/>For such entropy pairs, it is immediate to check that <lb/> ∂ <lb/>∂t <lb/> [ S(u ,x) + Σ(v ,x) ]+ <lb/> ∂ <lb/>∂x <lb/> [ S(u ,x) − Σ(v ,x) ] <lb/>= <lb/>1 <lb/> <lb/> ∂S <lb/>∂v <lb/> (u ,x) − <lb/> ∂Σ <lb/>∂v <lb/> (v ,x) <lb/> <lb/> h(v ,x) − u <lb/> <lb/> + <lb/> ∂S <lb/>∂x <lb/> (u ,x) − <lb/> ∂Σ <lb/>∂x <lb/> (v ,x). <lb/> <note place="headnote"> Benoˆ t Perthame, Nicolas Seguin, Magali Tournus <lb/></note> <page> 5 <lb/></page> Now, using (3.1)(ii), we can write <lb/> ∂ <lb/> ∂t <lb/> [S(u ,x)+ Σ(v ,x) ]+ <lb/> ∂ <lb/>∂x <lb/> [ S(u ,x) − Σ(v ,x) ] <lb/>= <lb/>1 <lb/> <lb/> ∂S <lb/>∂v <lb/> (u ,x) − <lb/> ∂S <lb/>∂v <lb/> (h(v ε ,x),x) <lb/> <lb/> h(v ,x) − u <lb/> <lb/> + <lb/> ∂S <lb/>∂x <lb/> (u ,x) − <lb/> ∂Σ <lb/>∂x <lb/> (v ,x) <lb/> ≤ <lb/> ∂S <lb/>∂x <lb/> (u ,x) − <lb/> ∂Σ <lb/>∂x <lb/> (v ,x), <lb/> because S is convex with respect to its first variable and thus <lb/> ∂S <lb/>∂v <lb/> is non decreasing <lb/>with respect to its first variable. <lb/>The shortcoming of Definition 3.1 is that the above right hand side is not always <lb/>well defined for u and v BV functions. Indeed, being given S(u,x), we compute <lb/> Σ(v,x) = <lb/> v <lb/> 0 <lb/> S <lb/> u <lb/> <lb/> h(¯ v,x),x <lb/> <lb/> d¯ v <lb/> and the expression for the x-derivative <lb/> ∂Σ <lb/>∂x <lb/> (v,x) = <lb/> v <lb/> 0 <lb/> <lb/> S <lb/> uu <lb/> <lb/> h(¯ v,x),x <lb/> <lb/> h x (¯ v,x)+ ··· <lb/> <lb/> d¯ v <lb/> does not make intrinsic sense. <lb/>However, this entropy inequality is enough to prove that equilibrium is reached. <lb/>Because the expressions <lb/> ∂S <lb/>∂x <lb/> (u ,x) and <lb/> ∂Σ <lb/>∂x <lb/> (v ,x) are bounded thanks to our assump-<lb/>tions on h, we may choose <lb/> S(u) = <lb/> u 2 <lb/> 2 <lb/> , <lb/> Σ(v,x) = <lb/> v <lb/> 0 <lb/> h(¯ v,x)d¯ v, <lb/> and using the entropy dissipation, with assumption (1.3), we find after integration of <lb/>equality (3) in (x,t), that for all T > 0 for some constant C(T ) which does not depend <lb/>on ε, it holds <lb/>1 <lb/> <lb/> T <lb/> 0 <lb/> L <lb/> 0 <lb/> <lb/> u (x,t) − h(v (x,t),x) <lb/> <lb/> dxdt ≤ C(T ). <lb/>Therefore, we arrive at the conclusion <lb/> Proposition 3.2. For u and v solutions of (1.1), we have the convergence <lb/> u − h(v ,x) −→ <lb/> −→0 <lb/> 0, <lb/> L 2 <lb/> <lb/> [0,L] × [0,T ] <lb/> <lb/> . <lb/> (3.2) <lb/>This convergence result toward equilibrium is an intermediate step in the proof of <lb/>Proposition 1.1. <lb/>We wish to go further and avoid the two terms containing x-derivatives. For this <lb/>goal, we define adapted (heterogeneous) entropies by imposing more restrictions on <lb/>the spatial dependence of the entropy pair. <lb/> Definition 3.3 (An adapted (heterogeneous) entropy family). The pair of con-<lb/>tinuous functions (S,Σ), is an adapted (heterogeneous) entropy for the system (1.1) <lb/> if it satisfies the conditions of Definition 3.1 and if <lb/> − <lb/> ∂S <lb/>∂x <lb/> (h(v,x),x)+ <lb/> ∂Σ <lb/>∂x <lb/> (v,x) = 0, <lb/> ∀v ≥ 0. <lb/>(3.3) <lb/> <page> 6 <lb/></page> <note place="headnote"> A simple detour for BV bounds in relaxations systems <lb/></note> An example of such a family of entropies parametrized by p ∈ R are the adapted <lb/>Kružkov entropies <lb/> S p (u,x) = |u − h(k p (x),x)|, <lb/> Σ p (v,x) = |v − k p (x)|, <lb/>(3.4) <lb/>where (k p ) p∈R is the family of stationary solutions of the limit equation (1.7), which <lb/>is equivalent to say <lb/> h(k p (x),x) − k p (x) = p. <lb/> With this choice of an entropy pair, the above entropy inequality then reduces to <lb/> ∂ <lb/> ∂t <lb/> [S p (u ,x)+ Σ p (v ,x) ]+ <lb/> ∂ <lb/>∂x <lb/> [ S p (u ,x) − Σ p (v ,x) ] ≤ 0. <lb/>As a consequence, thanks to the strong compactness proven in next part, in the <lb/>limit ε → 0, the quasilinear conservation law (1.7) comes with the family of adapted <lb/>Kružkov entropies. Indeed, if we define <lb/> ρ(x,t) := h(v(x,t),x)+ v(x,t), <lb/>A(ρ,t) := h(v(x,t),x) − v(x,t), <lb/> (3.5) <lb/>then, as in [1] and because h is increasing, the following entropy inequality holds in <lb/>the sense of distributions <lb/> ∂ <lb/>∂t <lb/> <lb/> <lb/>ρ(x,t) − <lb/> <lb/> k p (x)+ h(k p (x),x) <lb/> <lb/> <lb/> + <lb/> ∂ <lb/>∂x <lb/> <lb/> <lb/>A(ρ,x) − A <lb/> <lb/> k p (x)+ h(k p (x),x),x <lb/> <lb/> <lb/> ≤ 0. <lb/>(3.6) <lb/> 4. BV bounds <lb/> For well prepared initial conditions (1.5), we present here our method to prove <lb/>BV bounds for appropriate quantities and strong compactness for u and v , that are <lb/>points (ii), (iii) of Proposition 1.1. <lb/> 1st step. A bound on the time derivative at t = 0. Our first statement is <lb/> L <lb/> 0 <lb/> | <lb/> ∂u <lb/> ∂t <lb/> (x,0)|dx ≤ K 1 (u 0 ), <lb/> L <lb/> 0 <lb/> | <lb/> ∂v <lb/> ∂t <lb/> (x,0)|dx ≤ K 2 (v 0 ). <lb/>(4.1) <lb/>Indeed, because initial conditions are at equilibrium, we have <lb/> ∂v <lb/> ∂t <lb/> (x,0) − <lb/> ∂v <lb/> ∂x <lb/> (x,0) = 0. <lb/>We multiply this equality by sign <lb/> ∂ <lb/>∂t <lb/> v <lb/> <lb/> (x,0) and integrate over [0,L] to get <lb/> L <lb/> 0 <lb/> <lb/> <lb/> <lb/> ∂v <lb/> ∂t <lb/> (x,0) <lb/> <lb/> <lb/>dx = <lb/> L <lb/> 0 <lb/> <lb/> <lb/> <lb/> ∂v <lb/> ∂x <lb/> (x,0) <lb/> <lb/> <lb/>dx ≤ K 2 (v 0 ), <lb/>(4.2) <lb/>and the above inequality follows from assumption (1.4). This gives the first inequality <lb/>of estimates (4.1). The same argument applies for u . <lb/> 2nd step. The time BV estimate. To prove (2.1), we differentiate each line <lb/>of (1.1) with respect to time and we multiply it respectively by sign <lb/> ∂ <lb/>∂t <lb/> u <lb/> <lb/> and <lb/>sign <lb/> ∂ <lb/>∂t <lb/> v <lb/> <lb/> and integrate in x. Adding the two lines, we obtain <lb/> d <lb/>dt <lb/> L <lb/> 0 <lb/> [ | <lb/> ∂ <lb/>∂t <lb/> u | + | <lb/> ∂ <lb/>∂t <lb/> v | ](x,t)dx ≤ | <lb/> ∂ <lb/>∂t <lb/> u 0 |− | <lb/> ∂ <lb/>∂t <lb/> u (L,t)| + | <lb/> ∂ <lb/>∂t <lb/> v (L,t)|− | <lb/> ∂ <lb/>∂t <lb/> v (0,t)| <lb/>= − <lb/> 1 <lb/>2 <lb/> | <lb/> ∂ <lb/>∂t <lb/> u (L,t)|− | <lb/> ∂ <lb/>∂t <lb/> v (0,t)| ≤ 0, <lb/>(4.3) <lb/> <note place="headnote"> Benoˆ t Perthame, Nicolas Seguin, Magali Tournus <lb/></note> <page> 7 <lb/></page> which implies, using estimate (4.1), <lb/> L <lb/> 0 <lb/> [ | <lb/> ∂ <lb/> ∂t <lb/> u | + | <lb/> ∂ <lb/>∂t <lb/> v | ](x,t)dx ≤ <lb/> L <lb/> 0 <lb/> [ | <lb/> ∂ <lb/>∂t <lb/> u | + | <lb/> ∂ <lb/>∂t <lb/> v | ](x,0)dx ≤ K 1 (u 0 )+ K 2 (v 0 ). (4.4) <lb/> 3rd step. BV bound in x. We complete the proof Proposition 1.1 (ii). Because <lb/>of space dependence of h we cannot apply the same arguments for x-derivatives and <lb/>build a single BV quantity. We add the two lines of (1.1) and obtain <lb/> ∂ <lb/>∂x <lb/> (u − v ) <lb/> <lb/> (x,t) = − <lb/> ∂ <lb/>∂t <lb/> (u + v ) <lb/> <lb/> (x,t). <lb/>Using (2.1), we thus conclude that for all t ≥ 0 <lb/> L <lb/> 0 <lb/> <lb/> <lb/> <lb/> ∂ <lb/>∂x <lb/> (u − v ) <lb/> <lb/> <lb/>(x,t)dx ≤ K 1 (u 0 )+ K 2 (v 0 ). <lb/> 4th step. Compactness. <lb/> Therefore we can conclude that <lb/> <lb/> u − v <lb/> <lb/> is compact in L 1 <lb/> <lb/> [0,L] × [0,T ] <lb/> <lb/> . Now, <lb/>thanks to Proposition (3.2), <lb/> <lb/> h(v ,.) − u <lb/> <lb/> is compact in L 1 <lb/> <lb/> [0,L] × [0,T ] <lb/> <lb/> . A combi-<lb/>nation of these two last compact embeddings gives us that <lb/> h(v ,.) − v <lb/> is compact in L 1 <lb/> <lb/> [0,L] × [0,T ] <lb/> <lb/> . <lb/> (4.5) <lb/>Therefore, there is a function Q ∈ L ∞ (0,L) such that, after extraction of a subse-<lb/>quence, <lb/> h(v ,.) − v −→ <lb/> −→0 <lb/> Q(x), <lb/> and because v → h(v,x) − v is one-to-one, thanks to assumptions (1.2). In the same <lb/>way, we conclude that the sequence v converges. Gathering the informations above, <lb/>Proposition 1.1 (iii) is proved. <lb/>Then we can pass to the limit and obtain the last statement, Proposition 1.1 (iv). <lb/> 5. Extension to a more specific relaxation system <lb/> The method we have developed so far can be extended to a more realistic problem <lb/>arising in kidney physiology that motivated this study. The system introduced and <lb/>studied in [18] is written, for t ≥ 0 and x ∈ [0,L], <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> ∂C 1 <lb/> <lb/> ∂t <lb/> (x,t)+ <lb/> ∂C 1 <lb/> <lb/> ∂x <lb/> (x,t) = <lb/>1 <lb/>3 <lb/> <lb/> C 2 <lb/> (x,t)+ h(C 3 <lb/> (x,t),x)) − 2C 1 <lb/> (x,t) <lb/> <lb/> , <lb/> ∂C 2 <lb/> <lb/> ∂t <lb/> (x,t)+ <lb/> ∂C 2 <lb/> <lb/> ∂x <lb/> (x,t) = <lb/>1 <lb/>3 <lb/> <lb/> C 1 <lb/> (x,t)+ h(C 3 <lb/> (x,t),x)) − 2C 2 <lb/> (x,t) <lb/> <lb/> , <lb/> ∂C 3 <lb/> <lb/> ∂t <lb/> (x,t) − <lb/> ∂C 3 <lb/> <lb/> ∂x <lb/> (x,t) = <lb/>1 <lb/>3 <lb/> <lb/> C 1 <lb/> (x,t)+ C 2 <lb/> (x,t) − 2h(C 3 <lb/> (x,t),x)) <lb/> <lb/> , <lb/>C 1 <lb/> (0,t) = C 1 <lb/>0 , <lb/>C 2 <lb/> (0,t) = C 2 <lb/>0 , <lb/>C 3 <lb/> (L,t) = C 2 <lb/> (L,t), <lb/> t>0, <lb/> (5.1) <lb/>Again, we want to prove uniform BV bounds for a small parameter , which measures <lb/>the ratio between ionic exchanges and flow along the tubules. <lb/> <page> 8 <lb/></page> <note place="headnote"> A simple detour for BV bounds in relaxations systems <lb/></note> We make the same assumptions (1.2) (the condition 1 < β ≤ <lb/> ∂h <lb/>∂v <lb/> (v,x) can be re-<lb/>laxed to 1 ≤ <lb/> ∂h <lb/>∂v <lb/> (v,x)) and (1.3) and same hypotheses on the initial conditions, namely <lb/>they belong to BV and are at equilibrium which means <lb/> C 1 = C 2 = h(C 3 ,x). <lb/> Following (3.5), the conservative quantity ρ and the flux B are defined by <lb/> ρ(x,t) := 2h(C 3 (x,t),x)+ C 3 (x,t), B(ρ,t) := 2h(C 3 (x,t),x) − C 3 (x,t). <lb/>(5.2) <lb/>For p ∈ R, we define uniquely the steady state k p as <lb/> B(k p (x),x) = p. <lb/> (5.3) <lb/> Theorem 5.1 (Limit ε → 0). The functions C i <lb/> , i = 1, 2, 3 converge almost every-<lb/>where to bounded functions C i and the quantity ρ(x,t) is an entropy solution to <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> <lb/> ∂ <lb/>∂t <lb/>ρ(x,t)+ <lb/>∂ <lb/>∂x <lb/> B(ρ(x,t),x) = 0, <lb/> t>0, x ∈ [0,L], <lb/> ρ(0,t) = C 1 <lb/>0 + C 2 <lb/>0 + h −1 C 1 <lb/>0 + C 2 <lb/>0 <lb/> 2 <lb/> ,0 <lb/> <lb/> , <lb/>t>0, <lb/> ρ(x,0) = ρ 0 (x), <lb/> ρ 0 (x) := C 0 (x)+ 2h(C 0 (x),x), <lb/> x∈ [0,L]. <lb/>(5.4) <lb/>The entropy formulation of the conservation law, for adapted (heterogeneous) en-<lb/>tropies is written, following [1] again, <lb/> ∂ <lb/>∂t <lb/> <lb/> <lb/>ρ(x,t) − <lb/> <lb/> 2k p (x)+ h(k p (x),x) <lb/> <lb/> <lb/> + <lb/> ∂ <lb/>∂x <lb/> <lb/> <lb/>B(ρ,x) − B <lb/> <lb/> 2k p (x)+ h(k p (x),x),x <lb/> <lb/> <lb/> ≤ 0. <lb/>This family of inequalities is enough to prove uniqueness as in [14] (see [17] for details). <lb/>The boundary conditions are understood in the following sense <lb/> Boundary condition at x = 0. <lb/>For all k p such that k p (0) + 2h(k p (0),0) ∈ <lb/> I <lb/> <lb/> ρ(0,t),h −1 C 1 <lb/>0 + C 2 <lb/>0 <lb/> 2 <lb/> ,0 <lb/> <lb/> + C 1 <lb/>0 + C 2 <lb/>0 <lb/> <lb/> , we have <lb/>sign <lb/> <lb/> ρ(0,t) − h −1 ( <lb/> C 1 <lb/>0 + C 2 <lb/>0 <lb/> 2 <lb/> ,0) − C 1 <lb/>0 − C 2 <lb/>0 <lb/> <lb/> <lb/> B(ρ(0,t),0) − [2h(k p (0),0) − k p (0)] <lb/> <lb/> ≤ 0, <lb/>(5.5) <lb/> Boundary condition at x = L. <lb/> For all k p such that k p (L)+ 2h(k p (L),L) ∈ <lb/> I <lb/> <lb/> ρ(L,t),2w L + h −1 (w L ,L) <lb/> <lb/> , we have <lb/>sign <lb/> <lb/> ρ(L,t) − 2w L − h −1 (w L ,L) <lb/> <lb/> B(ρ(L,t),L) − [2h(k p (L),L) − k p (L)] <lb/> <lb/> ≥ 0, (5.6) <lb/>where w L (t) := lim <lb/> ε−→0 <lb/> C 1 <lb/> ε (L,t), <lb/>and where I(a,b) denotes the interval <lb/>(min(a,b),max(a,b)). <lb/> <note place="headnote"> Benoˆ t Perthame, Nicolas Seguin, Magali Tournus <lb/></note> <page> 9 <lb/></page> Following the arguments we gave for the 2 × 2 system, and that we do not repeat, <lb/>we can prove BV bounds in several steps <lb/>(i) C 1 <lb/> , C 2 <lb/> and C 3 <lb/> are bounded in L ∞ ((0,∞) × (0,L)), <lb/>(ii) C 2 <lb/> (x,t)+ h(C 3 <lb/> (x,t),x)) − 2C 1 <lb/> (x,t) −→ <lb/> →0 <lb/> 0, C 1 <lb/> (x,t)+ h(C 3 <lb/> (x,t),x)) − 2C 2 <lb/> (x,t) −→ <lb/> →0 <lb/> 0 in L 2 ((0,∞) × (0,L)), <lb/>(iii) <lb/> ∂C 1 <lb/> <lb/> ∂t , <lb/> ∂C 2 <lb/> <lb/> ∂t , <lb/> ∂C 3 <lb/> <lb/> ∂t are bounded in L ∞ <lb/> (0,∞;L 1 (0,L) <lb/> <lb/> , <lb/>(iv) <lb/> ∂C 1 <lb/> <lb/> ∂x + <lb/> ∂C 2 <lb/> <lb/> ∂x + <lb/> ∂C 3 <lb/> <lb/> ∂x is bounded in L ∞ <lb/> (0,∞;L 1 (0,L) <lb/> <lb/> . <lb/>These statements prove the convergence result in Theorem 5.1. <lb/> </body> <back> <div type="acknowledgement">Acknowledgement. Funding for this study was provided by the program EMER-<lb/>GENCE (EME 0918) of the Université Pierre et Marie Curie (Paris Univ. 6). <lb/></div> <listBibl> REFERENCES <lb/>[1] E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous <lb/>flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 253–265. <lb/>[2] C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with bound-<lb/>ary conditions, Comm. Partial Differential Equations, 4 (1979), pp. 1017–1034. <lb/>[3] F. Bouchut, Non linear stability of finite volume methods for hyperbolic conservation laws <lb/>and well balanced schemes for sources, Birkhaüser-Verlag, 2004. <lb/>[4] G. Q. Chen, C. D. Levermore, and T. P. Liu, Hyperbolic conservation laws with stiff relax-<lb/>ation terms and entropy, Comm. Pure Appl. Math., 47 (1994), pp. 787–830. <lb/>[5] C. M. Dafermos, Hyperbolic conservation laws in continuum physics, vol. 325 of Grundlehren <lb/>der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], <lb/>Springer-Verlag, Berlin, third ed., 2010. <lb/>[6] A.-L. Dalibard, Kinetic formulation for a parabolic conservation law. Application to homog-<lb/>enization, SIAM J. Math. Anal., 39 (2007), pp. 891–915 (electronic). <lb/>[7] R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., <lb/>88 (1985), pp. 223–270. <lb/>[8] F. Golse, P. L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the <lb/>solution of a transport equation, J. Funct. Anal., 76 (1988), pp. 110–125. <lb/>[9] F. James, Convergence results for some conservation laws with a reflux boundary condition <lb/>and a relaxation term arising in chemical engineering, SIAM J. Math. Anal., 29 (1998), <lb/>pp. 1200–1223 (electronic). <lb/>[10] S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary <lb/>space dimensions, Comm. Pure Appl. Math., 48 (1995), pp. 235–276. <lb/>[11] S. N. Kružkov, First order quasilinear equations with several independent variables., Mat. Sb. <lb/>(N.S.), 81 (123) (1970), pp. 228–255. <lb/>[12] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), <lb/>pp. 489–507. <lb/>[13] R. Natalini, Recent results on hyperbolic relaxation problems, in Analysis of systems of con-<lb/>servation laws (Aachen, 1997), vol. 99 of Chapman & Hall/CRC Monogr. Surv. Pure Appl. <lb/>Math., Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 128–198. <lb/>[14] R. Natalini and A. Terracina, Convergence of a relaxation approximation to a boundary <lb/>value problem for conservation laws, Comm. Partial Differential Equations, 26 (2001), <lb/>pp. 1235–1252. <lb/>[15] B. Perthame, Kinetic formulation of conservation laws, vol. 21 of Oxford Lecture Series in <lb/>Mathematics and its Applications, Oxford University Press, Oxford, 2002. <lb/>[16] M. Tournus, An asymptotic study to explain the role of active transport in models with coun-<lb/>tercurrent exchangers, SeMA Journal: Boletín de la Sociedad Española de Matemática <lb/>Aplicada, 59 (2012), pp. 19–35. <lb/>[17] <lb/>, Mod eles d'´ echanges ioniques dans le rein: théorie, analyse asymptotique et applications <lb/>numériques, PhD thesis, July 2013. <lb/>[18] M. Tournus, A. Edwards, N. Seguin, and B. Perthame, Analysis of a simplified model of <lb/>the urine concentration mechanism, Network Heterogeneous Media, 7 (2012), pp. 989 – <lb/>1018. <lb/>[19] M. Tournus, N. Seguin, B. Perthame, S. R. Thomas, and A. Edwards, A model of calcium <lb/> <page> 10 <lb/></page> <note place="headnote"> A simple detour for BV bounds in relaxations systems <lb/></note> transport along the rat nephron, American Journal of Physiology -Renal Physiology, 305 <lb/>(2013), pp. F979–F994. <lb/>[20] A. E. Tzavaras, Relative entropy in hyperbolic relaxation, Commun. Math. Sci., 3 (2005), <lb/>pp. 119–132. <lb/>[21] S. Y. Zhang and Y. G. Wang, Well-posedness and asymptotics for initial boundary value <lb/>problems of linear relaxation systems in one space variable, Z. Anal. Anwendungen, 23 <lb/>(2004), pp. 607–630. </listBibl> </back> </text> </tei>