RENORMALIZED ENTROPY SOLUTIONS FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS

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1 RENORMALIZED ENTROPY SOLUTIONS FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN Abstract. We prove the well posedness existence and uniqueness of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L 1 data. This paper complements the work by Chen and Perthame [19], who developed a pure L 1 theory based on the notion of kinetic solutions. 1. Introduction We consider the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L 1 data. This convection diffusion type problem is of the form 1.1 t u + divfu = au u + F, u0, x = u 0 x, where t, x 0, T R d ; T > 0 is fixed; div and are with respect to x R d ; and u = ut, x is the scalar unknown function that is sought. The initial and source data u 0 x and F t, x satisfy 1. u 0 L 1 R d, F L 1 0, T R d. The diffusion function au = a ij u is a symmetric d d matrix of the form 1.3 au = σuσu 0, σ L loc Rd K, 1 K d, and hence has entries a ij u = K σ ik uσ jk u, i, j = 1,..., d. Date: Revised version, June 9, Mathematics Subject Classification. 35K65, 35L65. Key words and phrases. degenerate parabolic equation, quasilinear, anisotropic diffusion, entropy solution, renormalized solution, uniqueness, existence. This work was supported by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT This work was done while MB visited the Centre of Mathematics for Applications CMA at the University of Oslo, Norway, and he is grateful for the hospitality. 1

2 M. BENDAHMANE AND K. H. KARLSEN The nonnegativity requirement in 1.3 means that for all u R a ij uλ i λ j 0, λ = λ 1,..., λ d R d. Finally, the convection flux fu is a vector valued function that satisfies 1.4 fu = f 1 u,..., f d u Lip loc R d. It is well known that 1.1 possesses discontinuous solutions and that weak solutions are not uniquely determined by their initial data the scalar conservation law is a special case of 1.1. Hence 1.1 must be interpreted in the sense of entropy solutions [3, 41, 4]. In recent years the isotropic diffusion case, for example the equation 1.5 t u + divfu = Au, Au = u 0 aξ dξ, 0 a L loc R, has received much attention, at least when the data are regular enough say L 1 L to ensure Au L. Various existence results for entropy solutions of 1.5 and 1.1 can be derived from the work by Vol pert and Hudjaev [4]. Some general uniqueness results for entropy solutions have been proved in the one-dimensional context by Wu and Yin [43] and Bénilan and Touré [6]. In the multi-dimensional context a general uniqueness result is more recent and was proved by Carrillo [15, 14] using Kružkov s doubling of variables device. Various extensions of his result can be found in [13, 7, 9, 31, 34, 35, 39], see also [17] for a different approach and [40] for a uniqueness proof for piecewise smooth weak solutions. Explicit continuous dependence on the nonlinearities estimates were proved in [0]. There are also several recent studies concerned with the convergence of numerical schemes for 1.5, see [5, 6, 30, 7, 36, 35,, 1]. In the literature just cited it is essential that the solutions u possess the regularity Au L. This excludes the possibility of imposing general L 1 data, since it is well known that in this case one cannot expect that much integrability see, e.g., the citations below on renormalized solutions. The general anisotropic diffusion case 1.1 is more delicate and was successfully solved only recently by Chen and Perthame [19]. Chen and Perthame introduced the notion of kinetic solutions and provided a wellposedness theory for 1.1 with L 1 data. Within their kinetic framework, explicit continuous dependence and error estimates for L 1 L entropy solutions were obtained in [18]. With the only assumption that the data belong to L 1, we cannot expect a solution of 1.1 to be more than L 1. Hence it is in general impossible to make distributional sense to 1.1 or its entropy formulation. In addition, as already mentioned above, we cannot expect au u to be square-integrable, which seems to be an essential condition for uniqueness. Both these problems were elegantly dealt with in [19] using the kinetic approach.

3 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 3 The purpose of the present paper is to offer an alternative pure L 1 well posedness theory for 1.1 based on a notion of renormalized entropy solutions and the classical Kružkov method [3]. The notion of renormalized solutions was introduced by DiPerna and Lions in the context of Boltzmann equations [3, ] see also [4]. This notion and a similar one was then adapted to nonlinear elliptic and parabolic equations with L 1 or measure data by various authors, see for example [10, 11, 38, 4, 3, 33, 7, 9, 1, 16, 8] the list is far from being complete. Bénilan, Carrillo, and Wittbold [5] introduced recently a notion of renormalized Kružkov entropy solutions for scalar conservation laws with L 1 data and proved the existence and uniqueness of such solutions. Their theory generalizes the Kružkov well posedness theory for L entropy solutions [3]. In passing, we mention that an alternative L 1 theory for scalar conservation laws has been developed by Perthame [37]. He has build up a theory around the notion of kinetic solutions, which is the notion that is generalized to 1.1 in [19]. Motivated by the above literature on renormalized solutions and [19], we introduce herein a notion of renormalized entropy solutions for 1.1 and prove its well posedness. Let us illustrate our notion of an L 1 solution on the isotropic diffusion equation 1.5 with initial data u t=0 = u 0 L 1. To this end, let T l : R R denote the truncation function at height l > 0 and let ζz = z 0 aξ dξ. A renormalized entropy solution of 1.5 is a function u L 0, T ; L 1 R d such that i ζt l u is square integrable on 0, T R d for any l > 0; ii for any convex C entropy-entropy flux triple η, q, r, with η bounded and q = η f, r = η a, there exists for any l > 0 a nonnegative bounded Radon measure µ l on 0, T R d, whose total mass tends to zero as l, such that 1.6 t ηt l u + divqt l u rt l u η T l u ζt l u + µ l t, x in D 0, T R d. Roughly speaking, 1.6 expresses the entropy condition satisfied by the truncated function T l u. Of course, if u is bounded by M, choosing l > M in 1.6 yields the usual entropy formulation for u. In other words, a bounded renormalized entropy solution is an entropy solution. However, in contrast to the usual entropy formulation, 1.6 makes sense also when u is merely L 1 and possibly unbounded. Intuitively the measure µ l should be supported on the set { u = l} and in particular carry information about the behavior of the energy on the set where u is large. The requirement is that the energy should be small for large values of u, that is, the total mass of the renormalization measure µ l should vanish as l. This is essential for proving uniqueness of a renormalized entropy solution. Being explicit, the existence proof reveals that µ l 0, T R d { u 0 >l} u 0 dx 0 as l. We prove existence of a renormalized entropy solution to 1.1 using an approximation procedure based on artificial viscosity [4] and bounded data. We derive a priori estimates and pass to the limit in the approximations.

4 4 M. BENDAHMANE AND K. H. KARLSEN Uniqueness of renormalized entropy solutions is proved by adapting the doubling of variables device due to Kružkov [3]. In the first order case, the uniqueness proof of Kružkov depends crucially on the fact that x Φx y + y Φx y = 0, Φ smooth function on R d, which allows for a cancellation of certain singular terms. The proof herein for the second order case relies in addition crucially on the following identity involving the Hessian matrices of Φx y: xx Φx y + xy Φx y + yy Φx y = 0, which, when used together with the parabolic dissipation terms like the one found in 1.6, allows for a cancellation of certain singular terms involving the second order operator in 1.1. Compared to [19], our uniqueness proof is new even in the case of bounded entropy solutions. The remaining part of this paper is organized as follows: In Section we introduce the notion of a renormalized entropy solution for 1.1 and state our main well posedness theorem. The proof of this theorem is given in Section 3 uniqueness and Section 4 existence.. Definitions and statement of main result We start by defining an entropy-entropy flux triple. Definition.1 entropy-entropy flux triple. For any convex C entropy function η : R R, the corresponding entropy fluxes are defined by q = q 1,..., q d : R R d and r = r ij : R R d d q u = η uf u, r u = η uau. We will refer to η, q, r as an entropy-entropy flux triple. For 1 k K and 1 i d, we let ζ ik u = u 0 σ ik ξ dξ, ζ k u = ζ 1k u,..., ζ dk u, and for any ψ CR u ζ ψ ik u = ψξσ ik ξ dξ, ζ ψ k u = ζ ψ 1k u,..., ζψ dk u. 0 Let us introduce the following set of vector fields: L 0, T ; L div; R d = { w = w 1,..., w d } d L 0, T R d : divw L 0, T R d. Following [19] we define an entropy solution as follows: Definition. entropy solution. An entropy solution of 1.1 is a measurable function u : 0, T R d R satisfying the following conditions:

5 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 5 D.1 u L 0, T ; L 1 R d L 0, T R d. D. For any k = 1,..., K, ζ k u L 0, T ; L div; R d. D.3 chain rule For any k = 1,..., K and ψ CR, divζ ψ k u = ψudivζ ku a.e. in 0, T R d and in L 0, T R d. D.4 Define the parabolic dissipation measure n u,ψ l t, x by K. n u,ψ t, x = ψut, x divζ k ut, x.1 For any entropy-entropy flux triple η, q, r, t ηu + xi q i u x i x j r ij u 0,T R d η uf n u,η in D 0, T R d, that is, for any 0 φ D0, T R d, ηu t φ + q i u xi φ + r ij u x i x j φ dx dt + η uf φ dx dt 0,T R d 0,T R d n u,η t, xφ dx dt. D.5 The initial condition is taken in the following strong L 1 sense: ess lim u, t u 0 L t 0 1 R d = 0. Note that, thanks to D.1 and ζ k 0 = 0 for k = 1,..., d, condition D. is satisfied once we know that divζ k u = xi ζ ik u L 0, T R d. An important contribution of Chen and Perthame [19] is to make explicit the point that the chain rule D.3 should be included in the definition of an entropy solution in the anisotropic diffusion case. They also note that D.3 is automatically fulfilled when au is a diagonal matrix, and can then be deleted from Definition.. This applies to the isotropic case 1.5. Uniqueness of an entropy solution in the sense of Definition. was proved in [19] using a kinetic formulation and regularization by convolution.

6 6 M. BENDAHMANE AND K. H. KARLSEN The present paper offers an alternative proof based on the more classical Kružkov method of doubling the variables [3]. Let us mention that D.4 implies the following Kružkov -type entropy condition for all c R:. t u c + ] xi [sign u c f i u f i c ] x i x j [sign u c A ij u A ij c sign u c F 0, A ij = a ij. In the isotropic case 1.5,. simplifies to [ ] t u c + div sign u c fu fc.3 Au Ac 0, c R. After Carrillo s work [15, 14], it is known that.3 implies uniqueness in the isotropic case 1.5. In the anisotropic case 1.1,. is not sufficient for uniqueness. Indeed, it is necessary to explicitly include the parabolic dissipation measure in the entropy condition, as is done in D.4. As we discussed in Section 1, for unbounded L 1 solutions Definition. is in general not meaningful. In [19] the authors use a notion of kinetic solutions to handle unbounded L 1 solutions. It is the purpose of this paper to use instead a notion renormalized entropy solutions. Before we can introduce this notion, let us recall the definition of the Lipschitz continuous truncation function T l : R R at height l > 0: l, u < l,.4 T l u = u, u l, l, u > l. We then suggest the following notion of an L 1 solution: Definition.3 renormalized entropy solution. A renormalized entropy solution of 1.1 is a measurable function u : 0, T R d R satisfying the following conditions: D.1 u L 0, T ; L 1 R d. D. For any k = 1,..., K, ζ k T l u L 0, T ; L div; R d, l > 0. D.3 renormalized chain rule For any k = 1,..., K and ψ CR, divζ ψ k T lu = ψt l udivζ k T l u, l > 0,

7 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 7 a.e. in 0, T R d and in L 0, T R d. D.4 For any l > 0, define the renormalized parabolic dissipation measure n u,ψ l t, x by K. t, x = ψt l ut, x divζ k T l ut, x.5 n u,ψ l For any l > 0 and any entropy-entropy flux triple η, q, r, with η bounded by K for some given K, there exists a nonnegative bounded Radon measure µ u,k l on 0, T R d such that t ηt l u + xi q i T l u x i x j r ij T l u 0,T R d η T l uf n u,η l that is, for any 0 φ D0, T R d, ηt l u t φ + q i T l u xi φ + + η T l uf t, xφ 0,T R d t, xφ dx dt n u,η l 0,T R d + µ u,k l in D 0, T R d, r ij T l u x i x j φ dx dt φ dµ u,k l 0,T R d t, x. D.5 The total mass of the renormalization measure µ u,k l l, that is, lim l µu,k l 0, T R d = 0. vanishes as D.6 The initial condition is taken in the following strong L 1 sense: ess lim u, t u 0 L t 0 1 R d = 0. Note that since T l u L 0, T R d, the integrals in.5 are all well defined. Moreover, if a renormalized entropy solution u belongs to L 0, T R d, then it is also an entropy solution in the sense Definition. simply let l in Definition.3. Our well posedness result is contained in the following theorem, which is proved in Section 3 uniqueness and Section 4 existence. Theorem.1 well posedness. Suppose 1., 1.3, and 1.4 hold. Then there exists a unique renormalized entropy solution u of 1.1.

8 8 M. BENDAHMANE AND K. H. KARLSEN It is worth while mentioning that Theorem.1 holds under merely local regularity assumptions on fu, au. Moreover, au can be discontinuous, which is of interest in some applications [13]. In the isotropic case it is not necessary to include the chain rule D.3 as a part of Definition.3 since it is then automatically fulfilled, as the following two lemmas show. Lemma.1 chain rule. Let 0 σ L loc R and ψ L loc R. Set βz = z 0 σξ dξ, β ψ z = z 0 ψξσξ dξ. Then, for any measurable function u such that βt l u H 1 loc Rd,.6 β ψ T l ux = ψt l ux βt l ux, l > 0, for a.e. x R d and in L loc Rd. Proof. As in [19],.6 is a consequence of standard Sobolev space theory see, e.g., [8]. For the sake of completeness, let us sketch a proof. Define the lower semicontinuous function β 1 : R R by Denote by E R the set β 1 w := inf {ξ R βξ = w}. E = { w R such that β 1 is discontinuous at w }. Since β is nondecreasing, E is at most countable. Introduce the function and the corresponding set v := βt l u H 1 loc Rd W 1,1 loc Rd E = { } x R d : vx E R d. Then the classical chain rule from Sobolev space theory gives xi Ψvx = Ψ vx xi vx, for a.e. x R d \ E, i = 1,..., d, where Ψ : R R is defined by Ψ v = ψβ 1 v, v R. However, on E the W 1,1 loc function v = vx is constant and hence from standard Sobolev space theory xi Ψv = xi v = 0, for a.e. x E, i = 1,..., d. By the definition of v and since β 1 βξ = ξ for all ξ, we obtain.6. Lemma.. Let u be a renormalized entropy solution to 1.1 with au being a diagonal matrix: au = diaga 1 u,..., a d u, 0 a i L loc R, i = 1,..., d.

9 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 9 For any ψ L loc R and i = 1,..., d, let u u ζ i u = ai ξ dξ, ζ ψ i u = ψξ a i ξ dξ. Then for i = 1,..., d 0.7 xi ζ ψ i T l ux, t = ψt l ux, t xi ζ i T l ux, t, l > 0, for a.e. x, t 0, T R d and in L 0, T R d. Proof. Since xi ζ i T l u L 0, T R d and u L 0, T ; L 1 R d see Definition.3, we have ζ i T l u L 0, T ; H 1 R d. Hence ζ i T l u, t H 1 R d for a.e. t 0, T. Then.7 follows from an application of Lemma Uniqueness of renormalized entropy solution For the uniqueness proof, we need a C 1 approximation of sign and a corresponding C approximation of the Kružkov entropy flux c, c R. Lemma 3.1. For ε > 0, set 1, ξ < ε, 3.1 sign ε ξ = sin π ε ξ, ξ ε, 1, ξ > ε. For each c R, the corresponding entropy function u η ε u, c = u c 0 sign ε ξ dξ is convex and belongs to C R with η ε C c R and η ε 1 so that the constant K appearing in Definition.3 is 1. Moreover, η ε is symmetric in the sense that η ε u, c = η ε c, u and for all u R η ε u, c ηu, c := u c as ε 0. For each c R and 1 i, j d, we define the entropy flux functions 3. u q ε i u, c = u r ε iju, c = u c u c sign ε ξ c f iξ dξ, sign ε ξ c A ijξ dξ, where A ij = a ij for 1 i, j d. Then as ε 0 qi ε u, c ηu, c := sign u c f i u f i c, 3.3 rij, ε c r ij u, c := sign u c A ij u A ij c, for 1 i, j d. Let q ε = q1 ε,..., qε d, rε = rij ε, and similarly for q, r.

10 10 M. BENDAHMANE AND K. H. KARLSEN The function 3.1 is taken from [17]. The Kružkov entropy fluxes q, r in 3.3 are symmetric. This is, however, not true for the entropy fluxes q ε, r ε, but the symmetry error goes to zero as ε 0, that is, q ε u, c q ε c, u 0, r ε u, c r ε c, u 0, u, c R, and this is sufficient for proving uniqueness. We are now ready to prove uniqueness of renormalized entropy solutions. Theorem 3.1 uniqueness. Suppose 1.3 and 1.4 hold. Let u and v be renormalized entropy solutions of 1.1 with data F L 1 0, T R d, u 0 L 1 R d and G L 1 0, T R d, v 0 L 1 R d, respectively. Then for a.e. t 0, T, 3.4 u, t v, t L 1 R d t u 0 v 0 L 1 R d + F s, Gs, L 1 R d ds. 0 In particular, the Cauchy problem 1.1 admits at most one renormalized entropy solution. Proof. We shall prove 3.4 using Kružkov s doubling of variables method [3]. When it is notationally convenient we drop the domain of integration. Let η ε, qi ε, rε ij be the entropy flux triple defined in Lemma 3.1, and denote by µ u l, µv l the corresponding renormalization measures. From the definition of an renormalized entropy solution for u = ut, x, 3.5 η ε T l u, c t φ + qi ε T l u, c xi φ + sign ε T l u c F t, xφ dx dt n u,sign ε c t, xφ dx dt rijt ε l u, c x i x j φ dx dt φ dµ u l t, x, c R, l > 0 and for 0 φ = φt, x D0, T R d. From the definition of renormalized entropy solution for u = us, y, 3.6 η ε T l v, c s φ + qi ε T l v, c yj φ + sign ε T l v c Gs, yφ dy ds n v,sign ε c s, yφ dy ds rijt ε l v, c y i y j φ dy ds φ dµ v l s, y, c R, l > 0 and for every 0 φ = φs, y D0, T R d.

11 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 11 Choose c = T l vs, y in 3.5 and integrate over s, y. Choose c = T l ut, x in 3.6 and integrate over t, x. Then adding the two resulting inequalities yields 3.7 η ε T l u, T l v t + s φ + + [qi ε T l u, T l v xi φ + qi ε T l v, T l u yi φ] [ r ε ijt l u, T l v x i x j φ + r ε ijt l v, T l u y i y j φ] dx dt dy ds sign ε T l u T l v F t, x Gs, y dx dt dy ds n u,sign ε c t, x + n v,sign ε c s, y φ dx dt dy ds φt, x, s, y dµ u l t, x dy ds φt, x, s, y dµ v l s, y dx dt, where φ = φt, x, s, y is any nonnegative function in D0, T R d. We introduce next a function 0 ω DR that satisfies ωσ = ω σ, ωσ = 0 for σ 1, and ωσ dσ = 1. For ρ > 0 and z R, let R ω ρ z = 1 ρ ω z ρ. We take our test function φ = φt, x, s, y to be of the form φt, x, s, y = ϕ t+s, x+y t s ωρ, x y, where ϕ D0, T R d, 0 ϕ 1, and ω t s ρ With this choice, we have 3.8 t + s φ = t + s ϕ t+s, x+y and, x y = ωρ x y t s ωρ, x y x + y φ = x + y ϕ t+s, x+y t s ωρ, x y. Introduce the Hessian matrices xx φ = x i x j φ, xy φ = x i y j φ, yy φ = y i y j φ. t s ωρ. Then one can check that the following crucial matrix equality holds: xx + xy + yy φ = xx + xy + yy ϕ t+s, x+y t s ωρ, x y.

12 1 M. BENDAHMANE AND K. H. KARLSEN Note that the two latter properties imply that for 1 i d 3.9 qi ε T l u, T l v xi φ + qi ε T l v, T l u yi φ = qi ε T l u, T l v xi + yi ϕ t+s, x+y t s ωρ, x y + [qi ε T l v, T l u qi ε T l u, T l v] yi φ and for 1 i, j d 3.10 rijt ε l u, T l v x i x j φ + rijt ε l v, T l u y i y j φ = r ε ijt l u, T l v r ε ijt l u, T l v x i y j φ xi xj + xi yj + yi yj ϕ t+s, x+y + [ r ε ijt l v, T l u r ε ijt l u, T l v ] y i y j φ. t s ωρ, x y We also have φt, x, s, y dµ u l t, x dy ds 3.11 ω t s ρ, x y dy ds dµ u l t, x µ u l 0, T Rd and similarly 3.1 φt, x, s, y dµ v l s, y dx dt µv l 0, T Rd. Insertion of into 3.7 gives 3.13 η ε T l u, T l v t + s ϕ t+s, x+y + + qi ε T l u, T l v xi + yi ϕ t+s, x+y rijt ε l u, T l v xi xj + xi yj + yi yj ϕ t+s, x+y ω t s ρ, x y dx dt dy ds sign ε T l u T l v F t, x Gs, y dx dt dy ds E 1 ε + E ε + E 3 ε µ v l 0, T Rd µ v l 0, T Rd,

13 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 13 where E j ε = I j ε dx dt dy ds, j = 1,, 3, with 0,T R d I 1 ε = n u,sign ε c t, x + n v,sign ε c s, y φ I ε = I 3 ε = rijt ε l u, T l v x i y j φ, [qi ε T l u, T l v qi ε T l v, T l u] yi φ + [ r ε ij T l u, T l v rijt ε l v, T l u ] y i y j φ. Thanks to Lemma 3.1, lim ε 0 E 3 ε = 0 and 3.14 lim E ε = ε 0 Our goal now is to show that r ij T l u, T l v x i y j φ dx dt dy ds lim ε 0 E 1 ε + lim ε 0 E ε 0. To this end, note first that, since sign ε 0, so that I 1 ε K sign ε T l u T l v div x ζ k T l udiv y ζ k T l vφ, E 1 ε K sign ε T l u T l v div x ζ k T l u div y ζ k T l vφ dx dt dy ds. Invoking the chain rule D.3 in Definition.3 we can do this since sign ε CR, we have for 1 k K 3.16 sign ε T l u T l v div y ζ k T l v = div y ζ sign ε T lu k T l v. If we now use 3.16, then we have E 1 ε K div x ζ k T l udiv y ζ sign ε T lu k T l vφ dx dt dy ds.

14 14 M. BENDAHMANE AND K. H. KARLSEN Integration by parts in y yields E 1 ε = K xi ζ ik T l u Tl v yj sign ε T l u ξ σ jk ξ dξ φ dx dt dy ds K T l u xi ζ ik T l u Tl v sign ε T l u ξ σ jk ξ dξ yj φ dx dt dy ds. T l u For 1 k K and 1 j d, define the function ψ ε jk : R R by Tl ψjk ε v η = sign ε η ξ σ jk ξ dξ. η Since sign ε CR and σ jk L loc R, we have ψε jk CR and the chain rule can therefore be used. Using the chain rule D.3 in Definition.3 and then doing integration by parts in x, we derive E 1 ε = = = K K K xi ζ ik T l uψjk ε T lu yj φ dx dt dy ds K Observe that for a.e. η R xi ζ ψε jk ik T lu yj φ dx dt dy ds Tl u xi ψjk ε ξσ ikξ dξ yj φ dx dt dy ds T l v Tl u ψjk ε ξσ ikξ dξ x i y j φ dx dt dy ds. T l v lim ψjk ε η = sign η T lv σ jk η, ε 0 so that by the dominated convergence theorem we have for a.e. t, x, s, y lim ε 0 Tl u T l v ψ ε jk ξσ ikξ dξ = Tl u T l v sign ξ T l v σ jk ξσ ik ξ dξ.

15 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 15 Hence, after another application of the dominated convergence theorem, 3.17 lim E 1 ε ε 0 = K Tl u sign ξ T l v σ jk ξσ ik ξ dξ T l v x i y j φ dx dt dy ds r ij T l u, T l v x i y j φ dx dt dy ds. Finally, adding 3.17 to 3.14 yields Summing up, sending ε 0 in 3.13 gives I time + I conv + I diff 3.18 t, x, s, yω t s ρ, x y dx dt dy ds sign T l u T l v F t, x Gs, y ω t s ρ, x y ϕ t+s, x+y dx dt dyds µ u l 0, T Rd µ v l 0, T Rd, where I time t, x, s, y = T l ut, x T l vs, y t + s ϕ t+s, x+y, I conv t, x, s, y = q i T l ut, x, T l vs, y xi + yi ϕ t+s, x+y, I diff t, x, s, y = r ij T l u, T l v xi xj + xi yj + yi yj ϕ t+s, x+y. Let us introduce the change of variables x = x+y, t = t+s, z = x y, τ = t s, which maps 0, T R d 0, T R d into Ω = R d R d { t, τ 0 t + τ T, 0 t τ T Observe that t + s ϕ t+s, x+y = ϕ t t, x, x + y ϕt, x, s, y = x ϕ t, x. This change of variables diagonalizes also the operator xx + xy + yy : xx + xy + yy ϕ t+s, x+y = x x ϕ t, x. Keeping in mind that x = x + z, y = x z, t = t + τ, s = t τ, }.

16 16 M. BENDAHMANE AND K. H. KARLSEN we may now estimate 3.18 as 3.19 I time + I conv I diff t, x, τ, zω ρ zδ ρ τ d t d x dτ dz Ω F t + τ, x + z G t τ, x z ω ρ zδ ρ τϕ t, x d x d t dτ dz where Ω µ u l 0, T Rd µ v l 0, T Rd, I time t, x, τ, z = T l u t + τ, x + z T l v t τ, x z ϕ t t, x, I conv t, x, τ, z = q i Tl u t + τ, x + z, T l v t τ, x z xi ϕ t, x, I diff t, x, τ, z = r ij Tl u t + τ, x + z, T l v t τ, x z x i x j ϕ. Sending ρ 0 in 3.19 gives 3.0 T l u T l v t ϕ + q i T l u, T l v xi ϕ + r ij T l u, T l v x i x j ϕ dx dt F t, x Gt, x ϕ dx dt µ u l 0, T Rd µ v l 0, T Rd. By standard arguments choosing a sequence of functions 0 ϕ 1 from D0, T R d that converges to 1 0,t R d and using the initial conditions for u, v in the sense of, say, D.6 in Definition.3, it follows from 3.0 that for a.e. t 0, T T l ut, x T l vt, x dx R d t 3.1 T l u 0 T l v 0 dx + F s, x Gs, x dx ds R d 0 R d + µ u l 0, T Rd + µ v l 0, T Rd. Equipped with D.5 in Definition.3 for u and v, sending l in 3.1 yields finally the L 1 contraction property 3.4.

17 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS Existence of renormalized entropy solution The purpose of this section is to prove the following theorem: Theorem 4.1 existence. Suppose 1., 1.3, and 1.4 hold. Then there exists at least one renormalized entropy solution u of 1.1. We divide the proof into two steps. Step 1 bounded data. Suppose the data u 0 and F are bounded and integrable functions. Repeating the proof in [19] we find that there exists a unique entropy solution u to 1.1 interpreted in the sense of Definition., and this entropy solution can be constructed by the vanishing viscosity method [4]. For us it remains to prove that this entropy solution is also a renormalized entropy solution in the sense of Definition.3. To this end, let u ρ be the unique classical say C 1, solution to the uniformly parabolic problem see [4] 4.1 t u ρ + divfu ρ = au ρ u ρ + ρ u ρ + F, ρ > 0, u ρ x, 0 = u 0 x. Equipped with the ρ-independent a priori estimates in [4], Chen and Perthame [19] prove 4. u ρ u a.e. and in C0, T ; L 1 R d as ρ 0, where u is the unique entropy solution to 1.1. For any C function S and qi S = S f i, rij S = S a ij for 1 i, j d, multiplying the equation in 4.1 by S u ρ yields 4.3 t Su ρ + xi qi S u ρ S u ρ F u ρ = x i x j riju S ρ ρ Su ρ n S ρ + m S ρ t, x, where the parabolic dissipation measure n S n,ρt, x is defined by K n S ρ t, x = xi ζik S ut, x. and the entropy dissipation measure m S ρ t, x is defined by m S ρ t, x = ρs u ρ u ρ. An easy approximation argument reveals that 4.3 continues to hold for any function S W, R. u Inserting Su = 1 l the well known estimate 0 T l ξξ into 4.3 and then sending l 0, we get 4.4 u ρ L 0,T ;L 1 R d u 0 L 1 R d + F L 1 0,T R d.

18 18 M. BENDAHMANE AND K. H. KARLSEN We need to derive some additional a priori estimates involving.4 that are independent of ρ and u 0 L R d, F L 0,T R d. Lemma 4.1. For any l > 0, we have K xi ζ ik T l u ρ + ρ T l u ρ dx dt C l, 0,T R d for some constant C l that is independent of ρ but not l. More precisely, C l = l u 0 L 1 R d + F L 1 0,T R d. Proof. Introduce the function Su = u 0 T l ξ dξ = Choosing this S in 4.3, we derive { u, u l, l u l, u > l. 4.5 T Su ρ T, x dx Su 0 x dx F u ρ T l u ρ dx dt R d R d 0 R d K = xi ζ ik T l u ρ + ρ T l u ρ dx dt. 0,T R d From the nonnegativity of S, it follows from 4.5 K xi ζ ik T l u ρ + ρ T l u ρ dx dt 0,T R 4.6 d T Su 0 x dx + F u ρ T l u ρ dx dt. R d 0 R d Since for all u R, T l u l and 0 Su l u, we deduce from 4.6 K xi ζ ik T l u ρ + ρ T l u ρ dx dt 0,T R d l u 0 L 1 R d + F L 1 0,T R d, which completes the proof of Lemma 4.1. Lemma 4.. For any l > 0 and any δ > 0, 1 K 4.7 xi ζ ik u ρ + ρ u ρ dx dt El, δ {l< u ρ <l+δ}

19 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 19 for some bounded function E on R + that is independent of n, ρ, δ and satisfies If the data u 0, F are bounded and lim El = 0. l 4.8 l > M := u 0 L R d + F L 0,T R d, then El = 0. Proof. Let us define the function S by S0 = 0 and 1, u < l δ, u l S u = 1 δ, l δ < u < l, δ T l+δu T l u = 0, l < u < l, u l δ, l < u < l + δ, 1, u > l + δ. Inserting this S into 4.3 gives 1 K xi ζ ik u ρ + ρ u ρ dx dt δ {l< u 4.9 ρ <l+δ} u 0 dx + F dx dt := El. { u 0 >l} { u ρ >l} Since u 0 L 1 R d, F L 1 0, T R d, and, thanks to 4.4, u ρ is uniformly in ρ bounded in L 1 0, T R d, we have El 0 as l. If the data u 0, F are bounded functions, then it is well known that u ρ L 0,T R d M, where M is defined in 4.8. We observe that if l > M, then Su 0 = 0 and S u ρ = 0. Hence we deduce El = 0. Let us choose a particular S = S η,h in 4.3 of the form This gives S η,h 0 = 0, S η,h = η h, η C R, η 0, η K, h C R, supph [ l, l] t S η,h u ρ + xi q S η,h i S η,h u ρf u ρ = u ρ x i x j r S η,h ij u ρ ρ S η,h u ρ n η h ρ + µ η h ρ t, x,

20 0 M. BENDAHMANE AND K. H. KARLSEN where µ η h ρ t, x := n η h ρ + m η h ρ t, x K = η u ρ h u ρ xi ζ ik u ρ + ρ u ρ. Let h l,δ : R R denote the function defined by h l,δ 0 = 0 and Clearly, 1, u < l, h l,δ u = l+δ u δ, l < u < l + δ, 0, u > l + δ h l,δ u T l u, h l,δ u 1 { u <l}, for any u R. The idea is to choose h = h n,l in 4.10 and then let δ 0. To this end, let us first define the Radon measure µ K l,ρ,δ on 0, T Rd by dµ K l,ρ,δ t, x := K K δ 1 {l< u ρ <l+δ} xi ζ ik u ρ + ρ u ρ dx dt, that is, for any Borel set E 0, T R d, µ K l,ρ,δ E = K δ Then, by Lemma 4., E {l< u ρ <l+δ} K xi ζ ik u ρ + ρ u ρ dx dt. Consequently, we may assume that µ K l,ρ,δ 0, T Rd El. 4.1 µ K l,ρ,δ µ K l,ρ µ K l,ρ in the sense of measures on 0, T R d as δ 0, µ K l in the sense of measures on 0, T R d ρ 0, for some nonnegative bounded Radon measure µ K l satisfying 4.13 µ K l 0, T R d El 0 as l.

21 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 1 For any 0 φ D0, T R d, thanks to 4.11 and the convexity of η, 4.14 lim δ 0 n 0,T η h l,δ ρ Rd t, xφ dx dt K = η u ρ 1 { uρ <l} xi ζ ik u ρ + ρ u ρ φ dx dt 0,T R d K η T l u ρ xi ζ ik T l u ρ φ dx dt. 0,T R d Again because of 4.11, it can be easily checked that as δ 0 recall q = η f and r = η a 4.15 S η,hl,δ u ηt l u, q S η,h l,δ u qt l u, S η,h l,δ u η T l u, r S η,h l,δ u rt l u, for any u R. Inserting h = h l,δ into 4.10 and using η K, 4.1, 4.14, 4.15 when sending δ 0, we get 4.16 t ηt l u ρ + xi q i T l u ρ x i x j r ij T l u ρ ρ ηt l u ρ η T l u ρ F K η T l u ρ xi ζ ik u ρ + µ K l,ρ in D 0, T R d. We now want to send ρ 0 in To pass to the limit in the first term on the right-hand side of 4.16 we employ a standard lower semicontinuity result found, e.g., in [1]. For any 0 φ D0, T R d we have 4.17 K lim inf η T l u ρ xi ζ ik T l u ρ φ dx dt ρ 0 0,T R d K η T l u xi ζ ik T l u φ dx dt. 0,T R d

22 M. BENDAHMANE AND K. H. KARLSEN Equipped with 4.17 and 4.1, passing to the limit ρ 0 in 4.16 yields ,T R d ηt l u t φ + q i T l u xi φ + r ij T l u x i x j φ dx dt + η T l uf φ dx dt 0,T R d K η T l u xi ζ ik T l u φ dx dt 0,T R d φ dµ K l t, x, φ D0, T R d, φ 0. 0,T R d It remains to prove that the chain rule D.3 in Definition.3 holds. For any ψ CR, the classical chain rule gives for k = 1,..., K xi ζ ψ ik T lu ρ = ψt l u ρ xi ζ ik T l u ρ, l > 0. As in [19], the proof is to observe that this equality continues to hold in the limit as ρ 0 since u ρ converges strongly and d x i ζ ik T l u ρ weakly. Step unbounded data. Suppose the data u 0 and F satisfy 1.. For n > 1, introduce the truncated data u 0,n = T n u 0 and F n = T n F. We have u 0,n u 0, F n F in L 1 as n. Thanks to the L 1 contraction property of the solution operator to 4.1, the following estimate holds for a.e. t 0, T : u n, t u n, t L 1 R d t u0,n u L 0,n 1 R d + F n s, F n s, L 1 R d ds 0, 0 as n, n. Hence {u n } n>1 is a Cauchy sequence in C0, T ; L 1 R d and has a limit point u. From Step 1 we know that each u n is a renormalized entropy solution of 1.1 with u 0 and F replaced by u 0,n and F n, respectively. Denote by µ K l,n the corresponding renormalization measure. Lemma 4.1 and 4.13 imply that the following n-independent a priori estimates hold for each l > 0: u n L 0,T ;L 1 R d u 0 L 1 R d + F L 1 0,T R d, K K divζ k T l u = xi ζ ik T l u n C l, µ K l,n 0, T Rd { u 0 >l} u 0 dx + { u n >l} F dx dt,

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25 QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 5 [35] A. Michel and J. Vovelle. Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods. Preprint, 00. [36] M. Ohlberger. A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. MAN Math. Model. Numer. Anal., 35: , 001. [37] B. Perthame. Kinetic formulation of conservation laws, volume 1 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, New York, 00. [38] J.-M. Rakotoson. Uniqueness of renormalized solutions in a T -set for the L 1 - data problem and the link between various formulations. Indiana Univ. Math. J., 43:685 70, [39] É. Rouvre and G. Gagneux. Solution forte entropique de lois scalaires hyperboliquesparaboliques dégénérées. C. R. Acad. Sci. Paris Sér. I Math., 397:599 60, [40] T. Tassa. Uniqueness of piecewise smooth weak solutions of multidimensional degenerate parabolic equations. J. Math. Anal. Applic., 10: , [41] A. I. Vol pert. The spaces BV and quasi-linear equations. Math. USSR Sbornik, :5 67, [4] A. I. Vol pert and S. I. Hudjaev. Cauchy s problem for degenerate second order quasilinear parabolic equations. Math. USSR Sbornik, 73: , [43] Z. Wu and J. Yin. Some properties of functions in BV x and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations. Northeastern Math. J., 54:395 4, Mostafa Bendahmane Department of Mathematics University of Bergen Johs. Brunsgt. 1 N 5008 Bergen, Norway address: mostafab@math.uio.no Kenneth Hvistendahl Karlsen Department of Mathematics University of Bergen Johs. Brunsgt. 1 N 5008 Bergen, Norway and Centre of Mathematics for Applications Department of Mathematics University of Oslo P.O. Box 1053, Blindern N 0316 Oslo, Norway address: kennethk@mi.uib.no URL: