Entropy Solution for Anisotropic Reaction-Diffusion-Advection Systems with L 1 Data

Transcription

1 Entropy Solution for Anisotropic Reaction-Diffusion-Advection Systems with L 1 Data Mostafa BENDAHMANE and Mazen SAAD Mathématiques Appliquées de Bordeaux Université Bordeaux I 351, cours de la Libération, F-3345 Talence Cedex mostafa@math.u-bordeaux.fr saad@math.u-bordeaux.fr Recibido: 12 de Noviembre de 23 Aceptado: 13 de Mayo de 24 ABSTRACT In this paper, we study the question of existence and uniqueness of entropy solutions for a system of nonlinear partial differential equations with general anisotropic diffusivity and transport effects, supplemented with no-flux boundary conditions, modeling the spread of an epidemic disease through a heterogeneous habitat. Key words: entropy solutions, uniqueness, anisotropic parabolic equations. 2 Mathematics Subject Classification: 35K57, 35K55, 92D3. 1. Introduction Let us first consider p = uv a simple population, where u = u(t) and v = v(t) are the respective densities of susceptible and infected individuals at time t. When no spatial consideration is involved the dynamics of the propagation of Feline Immunodeficiency Virus (F.I.V.) within a population of cats is governed by the following system of ordinary differential equations { u = σ(u, v) b(u v) (m k(u v))u, u() >, v = σ(u, v) αv (m k(u v))v, v() >, Rev. Mat. Complut. 25, 18; Núm. 1, ISSN:

2 where b is the (linear) natural birth rate, m is natural death rate and k > is a positive constant yielding a density dependent death rate δ(p) = mkp. For b m >, K p = b m k is the carrying capacity and p(t) K p as t. Let σ(u, v) be the incidence function, i.e. the recruitment of newly infected cats and α the disease induced death rate in the infected class. The incidence term is a mathematical expression describing the loss of individuals from the susceptible class and their entry into the latently infected class. Two commons of the incidence terms are proportionate mixing and mass action. In the case of mass action we have a term of the form σ(u, v) = σ 1 uv uv with σ 1 >, while a proportionate mixing term has the form σ(u, v) = σ 2 uv with σ 2 >. More details concerning the propagation of F.I.V. may be found in [15] and the references therein. Actually we shall be concerned with spatial densities and the total population of our subclass will be given by U(t) = u(t, x) dx and V (t) = v(t, x) dx where in R N (N 1) is a bounded domain representing the habitat under consideration. With this in mind the total population density p(t, x) is given by with total population p(t, x) = u(t, x) v(t, x), P (t) = p(t, x) dx. Here, u(t, x) and v(t, x) represent the spatial densities at time t and location x of susceptible and infectious individuals. We are led to consider spatially dependent birth rate b(t, x), death rate m(t, x) and the additional disease induced death rate in the infected class α(t, x). We denote by A i, K i and r i, for i = 1, 2 respectively the diffusivity terms, the transport field and the density dependent mortality rates terms (we see later the dependence of these functions on solutions). A prototype of a nonlinear system that governs the spreading of F.I.V. through a cat population in a heterogeneous spatial domain with seasonal variations and external supply is given by the following reaction-diffusion-advection system t u(t, x) div(a 1 (t, x, u(t, x)) u(t, x)k 1 (t, x)) r 1 (t, x, u, v) = σ(t, x, u, v) b(t, x)(u(t, x) v(t, x)) m(t, x)u(t, x) f(t, x), (1) t v(t, x) div(a 2 (t, x, v(t, x)) v(t, x)k 2 (t, x)) r 2 (t, x, u, v) = σ(t, x, u, v) (m(t, x) α(t, x))v(t, x) g(t, x); in = (, T ), together with no-flux boundary conditions on (, T ) (A 1 (t, x, u(t, x)) u(t, x)k 1 (t, x)) η(x) =, (A 2 (t, x, v(t, x)) v(t, x)k 2 (t, x)) η(x) =, K i (t, x) η(x) for i = 1, 2, (2) 25, 18; Núm. 1,

3 and initial distributions in u(, x) = u (x) and v(, x) = v (x), (3) where η denote the outward normal to on. Now, we precise the assumptions on the given functions which appears in the system (1). We confine ourselves to a model where the diffusivity A i, i = 1, 2, is a Carathéodory function in (, T ) R N whose components are a l,i for l = 1,..., N and i = 1, 2, satisfying for ξ R N : There exists p l > 1, a l,i (t, x, ξ) = β l,i (t, x) ξ l p l 2 ξ l. (4) Herein the nonnegative function β l,i is bounded on. We assume there exists a real positive constant a, such that for i = 1, 2 and for any ξ R N : A i (t, x, ξ) ξ a N ξ p l, a.e. (t, x). The transport vector K i (i = 1, 2) is bounded on and satisfies l=1 K i (L ( )) N and div(k i ) L ( ) for i = 1, 2. The functions m, b, and α are defined on with values in R and satisfy m, b, α L ( ). The density dependent mortality rates have the following form: { r 1 (t, x, u, v) = k 1 (t, x) u u v pu 1, r 2 (t, x, u, v) = k 2 (t, x) v u v pv 1 where p θ, for θ = u, v, satisfies p θ max 1 l N ( p l p l 1, p l) > 1, and the functions k i, i = 1, 2, defined on with values in R satisfy k i L ( ) and k i (t, x) k > a.e.(t, x) for i = 1, 2, 3. Last, σ : R R R R is measurable on, continuous with respect to u and v, a.e. in and satisfies a growth condition: there exists two bounded functions L, M : (, ) R N (, ), and s, s, R ( such ( that pl p N ), p u, p v ) 1 s < max and 1 l N p N 1 σ(t, x, u, v) L(t, x) ( u s v s ) M(t, x) a.e. (t, x), (5) 51 25, 18; Núm. 1, 49 67

4 where 1 p = 1 N N l=1 1 p l, and a nonnegativity condition σ(t, x,, v) = if v σ(t, x, u, ) if u σ(t, x, u, v) if u and v. (6) Before we discuss the concept of solution, we need to go into the functional setting especially the anisotropic Sobolev space. Set W 1,p,l () = { u W 1,1 u () x l L p () } the anisotropic Sobolev space, with u W 1,p,l () = u W 1,1 () u. x l We denote L p () L 1 () = {u L 1 (), u a.e. in }, and C 1 c ([, T ) ) the set of all C 1 -functions with compact support in [, T ). For given constant γ > we define the cut function T γ as the real-valued Lipschitz function T γ (z) = min(γ, max(z, γ)). By the Stampacchia Theorem ([1]), if u W 1,q () (q 1), we have T γ (u) = 1 { u <γ} u, where 1 { u <γ} denotes the characteristic function of a measurable set { u < γ} (, T ). We denote S γ (z) = Ψ γ (z) = z φ γ (τ) dτ. z T γ (τ) dτ, φ γ = T γ1 T γ and we set We note that T γ and φ γ are continuous Lipschitz functions, satisfying φ γ (z) 1 and Ψ γ (z) z for γ > and z R. The usual weak formulation of parabolic problems, in the case where the data are in L 1, does not ensure the uniqueness of the solution, some counterexamples are given in [17,2]. Then, for an isotropic parabolic equation with L 1 data, without nonlinear, reaction and advection terms, in [16] the author introduced a entropy formulation which allows to achieve existence and uniqueness. For the corresponding anisotropic parabolic equations with measure data or elliptic equations with L 1 data, existence of weak solutions is proved in [8] and [7] respectively. In [12], an existence result of entropy solutions to some parabolic equations is established. The data is considered in L 1 and no growth assumption is made on the lower-order term in divergence form. Another concept in terms of renormalized solutions permitting to ensure the uniqueness of the solution, can be found in [6, 18, 19]. Existence and uniqueness of renormalized solutions for a linear parabolic equation involving a first order term with free divergence coefficient is discussed in [14]. Also, in [9], existence and uniqueness 25, 18; Núm. 1,

5 of entropy solutions for linear parabolic equations involving th and 1 st order terms with L 1 data are proved. In this paper, we extend the results of [8, 16] to anisotropic reaction-diffusionadvection systems arising in population dynamics and modeling the propagation of Feline Immunodeficiency Virus. 2. Main results ( p N N1), l = 1,..., N. An entropy solu- Definition 2.1. Let 1 q l < p l p tion of (1) (3) is a couple (u, v) of nonnegative functions, with θ = u, v belonging to N l=1 Lq l (, T ; W 1,q l,l ()) L p θ (, T ; L p θ ()) C([, T ]; L 1 ()), such that σ(,, u, v) and r i (,, u, v), for i = 1, 2, belong to L 1 ( ), T γ (u) and T γ (v) belong to N l=1 Lp l (, T ; W 1,p l,l ()), and satisfying S γ (u ϕ)(t, x) dx T T S γ (u (x) ϕ(, x)) dx T t ϕ, T γ (u ϕ) dt T (A 1 (t, x, u) uk 1 (t, x)) T γ (u ϕ) dx dt T r 1 (t, x, u, v) T γ (u ϕ) dx dt σ(t, x, u, v)t γ (u ϕ) dx dt S γ (v ψ)(t, x) dx b (u v) T γ (u ϕ) dx dt T m u T γ (u ϕ) dx dt T S γ (v (x) ψ(, x)) dx T ft γ (u ϕ) dx dt, (7) t ψ, T γ (v ψ) dt T (A 2 (t, x, v) v K 2 (t, x)) T γ (v ψ) T r 2 (t, x, u, v) T γ (v ψ) dx dt T T (m α) v T γ (v ψ) dx dt σ(t, x, u, v)t γ (v ψ) dx dt T gt γ (v ψ) dx dt, (8) for all γ > and ϕ, ψ N l=1 Lp l (, T ; W 1,pl,l ()) L ( ) C([, T ]; L 1 ()) such that t φ, t ψ N l=1 L pl p l 1 (, T ; (W 1,pl,l ()) ). The results proved here are summarized in two theorems, the first one concerns the existence of entropy solutions and the second one establishes the uniqueness of these solutions , 18; Núm. 1, 49 67

6 Theorem 2.2. Assume that (4) (6) hold. Let u, v L 1 () and f, g L 1 ( ), then the system (1) (3) has an entropy solution. The uniqueness of entropy solution is established in the case where the incidence terms and the density dependent mortality rates are Lipschitz functions with respect to u and v. Theorem 2.3. Assuming that β(t, x) >, a.e. (t, x), β L ( ), such that σ(t, x, u 1, v 1 ) σ(t, x, u 2, v 2 ) β(t, x)( u 1 u 2 v 1 v 2 ) r i (t, x, u 1, v 1 ) r i (t, x, u 2, v 2 ) β(t, x)( u 1 u 2 v 1 v 2 ) for i = 1, 2, (9) then the entropy solution defined by (7) (8) is unique. Remark 2.4. Under Dirichlet boundary conditions, Theorem 2.2 and Theorem 2.3 remain valid by considering an adequate functional space. Precisely, the space W 1,ql,l () in Definition 2.1 is replaced by W 1,q l,l () = {u W 1,ql,l (), u = }. The plan of the paper is as follows. Section 3 is devoted to explain how the solution of system (1) is obtained and to precise its regularity. In the last two sections, we show existence and uniqueness of the entropy solution. 3. Approximate Solutions The method used in [3] for showing the existence of a solution for u, v L 1 () and f, g L 1 ( ) consists in : introducing a measurable may on, ˆσ continuous with respect to u and v, a.e. in, regularizing the following data f, g, u and v with nonnegative smooth sequences (f ε ) ε, (g ε ) ε (u,ε ) ε, and (v,ε ) ε. Then, classical results (see e.g. [13], [11]) provide the existence of a sequence u ε, v ε N l=1 Lp l (, T ; W 1,pl,l ()) L pmax ( ) C([, T ]; L 2 ()), p max = max(p u, p v ), with t u ε, t v ε N l=1 Lp l (, T ; (W 1,p l,l ()) ), of solutions of (1) (3) where u, v and f, g are replaced by u,ε, v,ε and f ε, g ε respectively, and σ is replaced by ˆσ, N l=1 Lp l (, T ; (W 1,p l,l ()) ) denotes the dual space of N l=1 Lp l (, T ; W 1,pl,l ()) with p l = p l p l 1. In order to obtain estimates on solutions independent of the parameter ε, the authors in [3] introduced a new parameter λ > satisfying λ max(b(t, x), b(t, x) m(t, x)div(k 1 (t, x)), div(k 2 (t, x))) a.e. (t, x). (1) 25, 18; Núm. 1,

7 We often omit the dependence of functions in (t, x) when no confusion is possible. We set u ε = e λt ũ ε and v ε = e λt ṽ ε ; then ũ ε and ṽ ε satisfy T T t ũ ε, ϕ dt T T à 1 (t, x, ũ ε ) ϕ dx dt r 1,λ (ũ ε, ṽ ε )ϕ dx dt T (λ m b)ũ ε ϕ dx dt T T ũ ε K 1 ϕ dx dt e λtˆσ(e λt ũ ε, e λt ṽ ε )ϕ dx dt bṽ ε ϕ dx dt = T e λt f ε ϕ dx dt, (11) T T t ṽ ε, ψ dt T à 2 (t, x, ṽ ε ) ψ dx dt ṽ ε K 2 ψ dx dt T T r 2,λ (ũ ε, ṽ ε )ψdx dt e λtˆσ(e λt ũ ε, e λt ṽ ε )ψ dx dt T T (λ m α)ṽ ε ψ dx dt = e λt g ε ψ dx dt, (12) for all ϕ, ψ N l=1 Lp l (, T ; W 1,p l,l ()) L ( ). Herein à i (t, x, ξ) = e λt A i (t, x, e λt ξ), for ξ R N, and r i,λ (t, x, ũ ε, ṽ ε ) = e λt r i (t, x, e λt ũ ε, e λt ṽ ε ). It is shown in [3] that the solutions satisfy (i) ũ ε, ṽ ε a.e. (t, x), (ii) c 1 >, c 2 > not depending on ε such that (iii) Let p N such that ũ ε ṽ ε L (,T ;L 1 ()) c 1, (13) r i,λ (ũ ε, ṽ ε ) L 1 ( ) σ(e λt ũ ε, e λt ṽ ε ) L 1 ( ) c 2 for i = 1, 2. (14) ( N N1, then for every 1 q l < p l p p z ε x l z ε Lq ( ) c 3, L q l (QT ) where z ε = ũ ε, ṽ ε and p satisfy 1 p = 1 N N l=1 1 p l. ) N N1, l = 1,..., N, c 3 > 55 25, 18; Núm. 1, 49 67

8 Let q = min 1 l N {q l }, then (ũ ε ) ε and (ṽ ε ) ε are bounded in L q (, T ; W 1,q ()), this implies that ( t ũ ε ) ε, ( t ṽ ε ) ε are bounded in L 1 (, T ; (W 1,q ()) )L 1 ( ); therefore, possibly at the cost of extracting subsequences still denoted (ũ ε ) ε and (ṽ ε ) ε (see Corollary 4 in [21]) we can assume that { ũ ε ũ strongly in L q ( ) and a.e. in, (15) ṽ ε ṽ strongly in L q ( ) and a.e. in. Finally, to treat the nonlinear terms and to prove the continuity in time of the solution, it is shown in [3] that σ(e λt ũ ε, e λt ṽ ε ) σ(e λt ũ, e λt ṽ) a.e. in and strongly in L 1 ( ), (16) r i,λ (ũ ε, ṽ ε ) r i,λ (ũ, ṽ), a.e. in and strongly in L 1 ( ), for i = 1, 2, (17) ( ũ ε (t, x), ṽ ε (t, x)) ( ũ(t, x), ṽ(t, x)) a.e. in (t, x) (, T ), (18) (ũ ε, ṽ ε ) (ũ, ṽ) strongly in (C(, T ; L 1 ())) 2. (19) Now, to complete the above estimates and the convergence results, we are concerned with the sequences (T γ (ũ ε )) ε and (T γ (ṽ ε )) ε. Proposition 3.1. Let (4) (6) hold. Then the sequences (T γ (ũ ε )) ε and (T γ (ṽ ε )) ε are uniformly bounded in N L p l (, T ; W 1,pl,l ()). (2) Proof. Let us choose ϕ = T γ (ũ ε ) as test function in (11), we have S γ (ũ ε )(T, x) dx S γ (ũ ε )(, x) dx à 1 (t, x, ũ ε ) T γ (ũ ε ) dx dt ũ ε K 1 T γ (ũ ε ) dx dt Q T e λt σ(e λt ũ ε, e λt ṽ ε )T γ (ũ ε ) dx dt (λ m b)ũ ε T γ (ũ ε ) dx dt Q T bṽ ε T γ (ũ ε ) dx dt r 1,λ (ũ ε, ṽ ε )T γ (ũ ε ) dx dt = e λt f ε T γ (ũ ε ) dx dt, (21) Using the positivity of solutions, the estimates (13), (14), the fact that T γ (ũ ε ) γ and S γ (ũ ε ) γ ũ ε, we deduce from (21) à 1 (t, x, ũ ε ) T γ (ũ ε ) dx dt ũ ε K 1 T γ (ũ ε ) dx dt (λ m b)ũ ε T γ (ũ ε ) dx dt Cγ (22) l=1 25, 18; Núm. 1,

9 where C is a positive constant independent of ε. Define F (z) = z yt γ(y)dy, then one gets F (z) zt γ (z). Using now the following equality, div(f (ũ ε )K 1 ) = div(k 1 )F (ũ ε ) K 1 F (ũ ε ), from (2) we have K 1 η on (, T ), thus ũ ε K 1 T γ (ũ ε ) dx dt (λ m b)ũ ε T γ (ũ ε ) dx dt (λ m b div(k 1 ))F (ũ ε ) dx dt. Consequently, from (22) we have which yields à 1 (t, x, ũ ε ) T γ (ũ ε )dx dt Cγ, (23) l=1 N e λt β l,1 (t, x) T γ (ũ ε ) x l dx dt Cγ. (24) pl It follows that (T γ (ũ ε )) ε is bounded sequence in N l=1 Lp l (, T ; W 1,p l,l ()). In the same way, one gets that the above estimate remains valid on solution ṽ ε and consequently the proof of estimate (2) is complete. 4. Existence of entropy solution Let us derive the entropy formulation for the regularized sequence (ũ ε ) ε. Let ϕ N l=1 Lp l (, T ; W 1,p l,l ()) L ( ) C([, T ]; L 1 ()) such that t ϕ N l=1 L p l p l 1 (, T ; (W 1,p l,l ()) ). Consider now φ = T γ (u ε ϕ) as test function in (11). Since u ε = e λt ũ ε and v ε = e λt ṽ ε, we have T S γ (u ε ϕ)(t, x) dx S γ (u ε ϕ)(, x) dx t ϕ, T γ (u ε ϕ) dt A 1 (t, x, u ε ) T γ (u ε ϕ) dx dt u ε K 1 T γ (u ε ϕ) dx dt Q T σ(u ε, v ε )T γ (u ε ϕ) dx dt mu ε T γ (u ε ϕ) dx dt 57 25, 18; Núm. 1, 49 67

10 b(u ε v ε )T γ (u ε ϕ) dx dt r 1 (u ε, v ε )T γ (u ε ϕ) dx dt = f ε T γ (u ε ϕ) dx dt, (25) Thus, using the fact that (u ε ) ε converges to u almost everywhere in, it follows from Proposition 3.1 that T γ (u ε ) T γ (u) weakly in N L p l (, T ; W 1,pl,l ()), (26) l=1 as ε goes to zero. Let us now study the limit for ε goes to of each term of equality (25). Since S γ is γ-lipschitz continuous and that u ε converges to u in C(, T, L 1 ()) (see (19)), when ε goes to, one gets and S γ (u ε ϕ)(t, x) dx S γ (u ε ϕ)(, x) dx S γ (u ϕ)(t, x) dx S γ (u ϕ)(, x) dx. We now pass to the limit in T tϕ, T γ (u ε ϕ) dt. Notice now that, setting k = ϕ L ( ), one has T γ (u ε ϕ) = T γ (T γk (u ε ) ϕ) and T γ (u ϕ) = T γ (T γk (u) ϕ). Since for l = 1,..., N (T γ (T γk (u ε ) ϕ) = 1 { Tγk (u x ε) ϕ γ} (T γk (u ε ) ϕ), l x l the weak convergence of T γk (u ε ) from (26) and T γk (u ε ) converges almost everywhere, moreover t ϕ in N l=1 Lp l (, T ; (W 1,p l,l ()) ). Hence T which is equivalent to t ϕ, T γ (T γk (u ε ) ϕ) dt T t ϕ, T γ (u ε ϕ) dt T T t ϕ, T γ (T γk (u) ϕ) dt, t ϕ, T γ (u ϕ) dt. 25, 18; Núm. 1,

11 One observes that A 1 (t, x, u ε ) T γ (u ε ϕ) dx dt = A 1 (t, x, T γk (u ε )) T γk (u ε )1 { Tγk (u) ϕ γ } dx dt A 1 (t, x, T γk (u ε )) ϕ1 { Tγk (u) ϕ γ } dx dt. Indeed, from the Proposition 3.1, the convergence (18), and the assumption (4), we deduce, when ε goes to, that a l,1 (t, x, T γ (u ε )) a l,1 (t, x, T γ (u)) weakly in L p l p l 1 ( ), for l = 1,..., N, where a l,1 are components of A 1. Using the dominated convergence theorem and the convergence of ϕ x l 1 { Tγk (u ε) ϕ γ} in L p l ( ), for each l = 1...N, we deduce A 1 (t, x, T γk (u ε )) ϕ1 { Tγk (u ε) ϕ γ } Thus from Fatou s Lemma, we have A 1 (t, x, T γk (u)) ϕ1 { Tγk (u) ϕ γ }. A 1 (t, x, T γk (u)) T γk (u)1 { Tγk (u) ϕ γ } dx dt lim inf A 1 (t, x, T γk (u ε ) T γk (u ε )1 { Tγk (u ε ε) ϕ γ } dx dt. Next, u ε K 1 T γ (u ε ϕ)dx dt = T γk (u ε )K 1 (T γk (u ε ) ϕ)1 { Tγk (u ε) ϕ γ}dx dt. Since (u ε ) ε converges to u almost everywhere in and convergence result (26), one gets T γk (u ε )K 1 (T γk (u ε ) ϕ)1 { Tγk (u ε) ϕ γ } dx dt T γk (u)k 1 (T γk (u) ϕ)1 { Tγk (u) ϕ γ } dx dt = uk 1 T γ (u ϕ) dx dt, 59 25, 18; Núm. 1, 49 67

12 when ε goes to zero. We complete the existence part of Theorem 2.2 by using the dominated convergence theorem to obtain that T γ (u ε ϕ) converges to T γ (u ϕ) weak in L ( ) and from the strong convergence results (15), (16), (17) in L 1 ( ), hence, when ε goes to, σ(u ε, v ε )T γ (u ε ϕ) dx dt σ(u, v)t γ (u ϕ) dx dt, Q T ((m b)u ε bv ε )T γ (u ε ϕ) dx dt ((m b)u bv)t γ (u ϕ) dx dt, Q T r 1 (u ε, v ε )T γ (u ε ϕ) dx dt r 1 (u, v)t γ (u ε ϕ) dx dt, Q T f ε T γ (u ε ϕ) dx dt ft γ (u ϕ) dx dt. Now passing to the limit as ε goes to zero on the formulation (25) we obtain that the limit u satisfies (7). In the same way, one gets v entropy solution. Remark 4.1. Note that, one can prove, when ε tends to, and T T A 1 (t, x, T γ (u ε )) T γ (u ε ) dx dt A 2 (t, x, T γ (v ε )) T γ (v ε ) dx dt T T A 1 (t, x, T γ (u)) T γ (u) dx dt, A 2 (t, x, T γ (v)) T γ (v) dx dt, T γ (z ε ) T γ (z) strongly in L p l (, T ; W 1,p l,l ()) for l = 1,..., N, where z ε = u ε, v ε and z = u, v, (see [1]). Then, the inequalities in (7) and (8) are equalities, however the inequality in the entropy formulation is sufficient to have uniqueness. 5. Uniqueness of entropy solution In this section, we study the uniqueness question of the entropy solutions constructed in this paper. The method we adopt, to prove Theorem 2.3, is rather close to those introduced in [4, 9, 16]. However, new difficulties arise essentially related to the influence of the transport terms and nonlinear terms. For that, we prove the uniqueness 25, 18; Núm. 1,

13 of the solutions (ũ, ṽ) defined by ũ = e λt u and ṽ = e λt v, where (u, v) is the entropy solution defined by (7)-(8) and λ is defined by (1). We often write and F (u, v) = e λt (f σ(e λt u, e λt v) r 1 (e λt u, e λt v)) b(u v) mu, G(u, v) = e λt (g σ(e λt u, e λt v) r 2 (e λt u, e λt v)) (m α)v. Recall that (ũ, ṽ) satisfies the following relations S γ (ũ ϕ)(t, x) dx S γ (u (x) ϕ(, x)) dx T T t ϕ, T γ (ũ ϕ) dt S γ (ṽ ψ)(t, x) dx T T λũt γ (ũ ϕ) dx dt t ψ, T γ (ṽ ψ) dt T (Ã1(t, x, ũ) ũk 1 (t, x)) T γ (ũ ϕ) dx dt T S γ (v (x) ψ(, x)) dx F (ũ, ṽ)t γ (ũ ϕ) dx dt, (27) (Ã2(t, x, ṽ) ṽ K 2 (t, x)) T γ (ṽ ψ) dx dt T T λṽt γ (ṽ ψ) dx dt G(ũ, ṽ)t γ (ṽ ψ) dx dt, (28) for all γ > and the test functions ϕ and ψ are defined in the definition 2.1. The proof of Theorem 2.3 is divided in two steps. The first step consists in establishing Lemma 5.1. This lemma proves that entropy solutions are limit of solutions obtained by approximation. And the second step is devoted to prove the uniqueness of entropy solutions in general. This result is a consequence of this lemma. Lemma 5.1. Let (ũ 2, ṽ 2 ) be the entropy solution solution of system defined by (27) (28). Let (ũ 1, ṽ 1 ) be the limit of solution obtained by approximation of (ũ 1,ε ) ε, (ṽ 1,ε ) ε solution of equations (11) and (12). Then (ũ 2, ṽ 2 ) = (ũ 1, ṽ 1 ) almost everywhere in (, T ). Proof. According to [17], we introduce the functions Th n and Rn h in C2 (R, R), such that for h > 1 n, (T n h ) (s) = if s h, (T n h ) (s) = 1 if s h 1 n, (T n h ) (s) 1 for all s, R n h() =, (R n h) (s) = 1 (T n h ) and R n h( s) = R n h(s), for all s , 18; Núm. 1, 49 67

14 Note that T n h (ṽ 1,ε) and (T n h ) (ṽ 1,ε ) are in N l=1 Lp l (, T ; W 1,p l,l ()) L ( ). Let us substitute ψ = T n h (ṽ 1,ε) in equation (28) where ũ and ṽ are replaced respectively by ũ 2 and ṽ 2. We have S γ (ṽ 2 Th n (ṽ 1,ε ))(t, x) dx t S γ (v (x) T n h (v,ε (x))) dx t t t ṽ 1,ε, (T n h ) (ṽ 1,ε )T γ (ṽ 2 T n h (ṽ 1,ε )) dτ Ã 2 (t, x, ṽ 2 ) T γ (ṽ 2 T n h (ṽ 1,ε )) dx dτ ṽ 2 K 2 T γ (ṽ 2 T n h (ṽ 1,ε )) dx dτ t t λ ṽ 2 T γ (ṽ 2 T n h (ṽ 1,ε )) dx dτ G(ũ 2, ṽ 2 )T γ (ṽ 2 T n h (ṽ 1,ε )) dx dτ. (29) Hence, let ψ N l=1 Lp l (, T ; W 1,p l,l ()) L ( ), and take (T n h ) (ṽ 1,ε )ψ as test function in (12), where ũ ε and ṽ ε are replaced respectively by ũ 1,ε and ṽ 1,ε. we get t t t ṽ 1,ε, (T n h ) (ṽ 1,ε )ψ dτ t t (T n h ) (ṽ 1,ε )A 2 (t, x, ṽ 1,ε ) ψ dx dτ (T n h ) (ṽ 1,ε ) ṽ 1,ε K 2 ψ dxdτ (T n h ) (ṽ 1,ε )ψã2(t, x, ṽ 1,ε ) ṽ 1,ε dx dτ = t t t (T n h ) (ṽ 1,ε )ψ ṽ 1,ε K 2 ṽ 1,ε dx dτ λ ṽ 1,ε (T n h ) (ṽ 1,ε )ψ dx dτ G(ũ 1,ε, ṽ 1,ε )(T n h ) (ṽ 1,ε )ψ dx dτ. Now, choosing ψ = T γ (ṽ 2 T n h (ṽ 1,ε)) in the above inequality, the entropy formula- 25, 18; Núm. 1,

15 tion (29) is equivalent to S γ (ṽ 2 Th n (ṽ 1,ε ))(t, x) dx S γ (v (x) Th n (v,ε (x))) dx t (Th n ) (ṽ 1,ε )T γ (ṽ 2 Th n (ṽ 1,ε )) ( ) Ã 2 (t, x, ṽ 1,ε ) ṽ 1,ε K 2 ṽ1,ε dx dτ t (T n h ) (ṽ 1,ε ) ( Ã 2 (t, x, ṽ 1,ε ) ṽ 1,ε K 2 ) Tγ (ṽ 2 T n h (ṽ 1,ε )) dx dτ t t t (Ã2 (t, x, ṽ 2 ) ṽ 2 K 2 ) Tγ (ṽ 2 T n h (ṽ 1,ε )) dx dτ λ ( ṽ 2 ṽ 1,ε (T n h ) (ṽ 1,ε ) ) T γ (ṽ 2 T n h (ṽ 1,ε )) dx dτ ( G(ũ2, ṽ 2 ) G(ũ 1,ε, ṽ 1,ε )(T n h ) (ṽ 1,ε ) ) T γ (ṽ 2 T n h (ṽ 1,ε )) dx dτ. (3) Let us denote the six integrals of the left hand side as L 1 to L 6 and the integral of the right hand side as L 7. In order to obtain an estimate on L 3, we take ψ = (Rh n) (ṽ 1,ε ) in equation (12). We have Rh(ṽ n 1,ε (t, x)) dx Rh(v n,ε (x)) dx t t (R n h) (ṽ 1,ε ) ( Ã 2 (t, x, ṽ 1,ε ) ṽ 1,ε K 2 ) ṽ1,ε dx dτ λṽ 1,ε (R n h) (ṽ 1,ε )dx dτ = t G(ũ 1,ε, ṽ 1,ε )(R n h) (ṽ 1,ε ) dx dτ. (31) Note that from the choice of λ in (1) and the fact that K 2 (t, x) η(x) on (, T ), we have t t λṽ 1,ε (Rh) n (ṽ 1,ε ) dx dτ ṽ 1,ε K 2 (Rh) n (ṽ 1,ε ) dx dτ. (32) We deduce from the definition of the function Rh n and estimate (32) that t L 3 γ (Th n ) (ṽ 1,ε )Ã2(t, x, ṽ 1,ε ) ṽ 1,ε (Th n ) (ṽ 1,ε )ṽ 1,ε K 2 ṽ 1,ε dx dτ T = γ γ T (R n h) (ṽ 1,ε )Ã2(t, x, ṽ 1,ε ) ṽ 1,ε (R n h) (ṽ 1,ε )ṽ 1,ε K 2 ṽ 1,ε dx dτ (R n h) (ṽ 1,ε )Ã2(t, x, ṽ 1,ε ) ṽ 1,ε λṽ 1,ε (R n h) (ṽ 1,ε ) T (Rh) n (ṽ 1,ε )ṽ 1,ε K 2 ṽ 1,ε dx dτ γλ ṽ 1,ε (Rh) n (ṽ 1,ε ) dx dτ, 63 25, 18; Núm. 1, 49 67

16 and using the equation (31) one gets ( T L 3 Cγ G(ũ 1,ε, ṽ 1,ε ) 1 { ṽ1,ε h 1 n } dx dτ ) v,ε 1 { v,ε h 1 n } dx, T Cγ ṽ 1,ε 1 { ṽ1,ε h 1 n } dx dτ, (33) where C denotes a positive constant independent of ε. Using the Dominated Convergence Theorem and estimate (33), it follows after letting n go to in estimate (3) that S γ (ṽ 2 T h (ṽ 1,ε (t, x))) dx S γ (v T h (v,ε (x))) dx t (Ã2 (t, x, ṽ 2 ) Ã2(t, x, T h (ṽ 1,ε )) ) T γ (ṽ 2 T h (ṽ 1,ε )) dx dτ t t t (ṽ 2 ṽ 1,ε (T h ) (ṽ 1,ε ))K 2 T γ (ṽ 2 T h (ṽ 1,ε )) dx dτ λ(ṽ 2 ṽ 1,ε (T n h ) (ṽ 1,ε ))T γ (ṽ 2 T h (ṽ 1,ε )) dx dτ (G(ũ 2, ṽ 2 ) G(ũ 1,ε, ṽ 1,ε )(T h ) (ṽ 1,ε ))T γ (ṽ 2 T h (ṽ 1,ε )) dx dτ ( t ) Cγ G(ũ 1,ε, ṽ 1ε ) 1 { ṽ1,ε h } dx dτ v,ε 1 { v,ε h } dx Cγ T Thanks to the monotonicity of Ã2, we have by Fatou s lemma t (Ã2 (t, x, ṽ 2 ) Ã2(t, x, T h (ṽ 1 )) ) T γ (ṽ 2 T h (ṽ 1 )) dx dτ lim inf ε T ṽ 1,ε 1 { ṽ1,ε h } dx. (Ã2 (t, x, ṽ 2 ) A 1 (t, x, T h (ṽ 1,ε )) ) T γ (ṽ 2 T h (ṽ 1,ε )) dxdτ, and the properties of the sequences ṽ 1,ε and ũ 1,ε allow to pass to the limit as ε. It remains to consider h in the following relation: S γ (ṽ 2 T h (ṽ 1 (t, x))) dx S γ (v T h (v (x))) dx t t (Ã2 (t, x, ṽ 2 ) Ã2(t, x, T h (ṽ 1 )) ) T γ (ṽ 2 T h (ṽ 1 )) dx dτ (ṽ 2 ṽ 1 (T h ) (ṽ 1 )) K 2 T γ (ṽ 2 T h (ṽ 1 )) dx dτ 25, 18; Núm. 1,

17 t t λ (ṽ 2 ṽ 1 (T h ) (ṽ 1 )) T γ (ṽ 2 T h (ṽ 1 )) dx dτ (G(ũ 2, ṽ 2 ) G(ũ 1, ṽ 1 )(T h ) (ṽ 1 ))T γ (ṽ 2 T h (ṽ 1 )) dx dτ where R(h) stands for t R(h) = G(u 1, v 1 ) 1 { u1 h } dx dτ which goes to as h. Now, let us write γr(h), (34) T v 1 { v h } dx v 1 1 { v1 h } dx, G(ũ 2, ṽ 2 ) G(ũ 1, ṽ 1 )(T h ) (ṽ 1 ) = G(ũ 1, ṽ 1 )(1 (T h ) (ṽ 1 )) (G(ũ 2, ṽ 2 ) G(ũ 1, ṽ 1 )). We have (1 (T h ) (ṽ 1 )) as h, then the term t G(ũ 1, ṽ 1 )(1 (T h ) (ṽ 1 ))T γ (ṽ 2 T h (ṽ 1 )) dx dτ can be included in the general expression γr(h) which tends to zero as h, and using the fact that T γ (.) γ and the function G is Lipschitz, we have t t (G(ũ 2, ṽ 2 ) G(ũ 1, ṽ 1 ))T γ (ṽ 2 T h (ṽ 1 )) dx dτ Cγ ( ũ 2 ũ 1 ṽ 2 ṽ 1 ) dx dτ. In the classical way, by Lebesgue s theorem and Fatou s lemma, letting h going to, we deduce from (34) S γ (ṽ 2 ṽ 1 )(t, x)dx t t (ṽ 2 ṽ 1 )K 2 T γ (ṽ 2 ṽ 1 ) dx dτ (Ã2 (t, x, ṽ 2 ) Ã2(t, x, ṽ 1 ) ) T γ (ṽ 2 ṽ 1 ) dx dτ Cγ t t λ(ṽ 2 ṽ 1 )T γ (ṽ 2 ṽ 1 ) dx dτ ( ũ 2 ũ 1 ṽ 2 ṽ 1 ) dx dτ. Then by the choice of λ in (1), the coercivity of Ã2 and letting γ going to, we have t ṽ 2 ṽ 1 (t, x) dx C ( ũ 2 ũ 1 ṽ 2 ṽ 1 )(t, x) dx dτ. for every t (, T ). In the same way, we obtain for every t (, T ) t ũ 2 ũ 1 (t, x)dx C ( ũ 2 ũ 1 ṽ 2 ṽ 1 )(t, x) dx dτ , 18; Núm. 1, 49 67

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