WELLPOSEDNESS FOR NONISOTROPIC DEGENERATE PARABOLICHYPERBOLIC EQUATIONS


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1 WELLPOSEDNESS FOR NONISOTROPIC DEGENERATE PARABOLICHYPERBOLIC EQUATIONS GUIQIANG CHEN AND BENOÎT PERTHAME In Memory of Ph. Bénilan Abstract. We develop a wellposedness theory for solutions in L 1 to the Cauchy problem of general degenerate parabolichyperbolic equations with nonisotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that L 1 is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in L 1, especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case. Résumé Nous développons une théorie d existence et unicité pour les solutions L 1 seulement du problème de Cauchy pour un problème de Cauchy hyperboliqueparabolique avec diffusion nonisotrope générale. Des notions de formulations entropique et cinétique sont introduites qui incorporent un nouvel ingrédient, une condition de type dérivation composée, qui montre la différence fondamentale avec le cas d une diffusion isotrope. L avantage de la notion de solution cinétique est de travailler directement dans l espace naturel L 1. Keywords: Kinetic solutions, entropy solutions, kinetic formulation, degenerate parabolic equations, convectiondiffusion, nonisotropic diffusion, stability, existence, wellposedness Subj. numbers: 35K65, 35K1, 35B3, 35D5 1. Introduction and Main Theorems Consider the Cauchy problem of a general nonlinear degenerate parabolichyperbolic equation of secondorder: 1.1 t u + fu= Au u, x, t, 1
2 1.2 u t= = u L 1, where f :IR satisfies 1.3 a :=f L loc IR; IRd, and the d d matrix Au =a ij u is symmetric, nonnegative, and locally bounded so that we can always write 1.4 a ij u = σ ik uσ jk u, σ ik L loc IR, and σ ik u is its square root matrix, in which the structure appears more naturally with the additional index K that can be thought to be the maximal rank of the matrix. Equation 1.1 and its variants model degenerate diffusionconvection motions of ideal fluids and arise in a wide variety of important applications, including two phase flows in porous media cf. [8] and the references cited therein and sedimentationconsolidation processes cf. [5] and the references cited therein. Since its importance in applications, there is a large literature for the design and analysis of various numerical methods to calculate solutions of 1.1 and its variants; see [8, 14, 13, 1, 19] and the references cited therein, for which a wellposedness theory for 1.1 is in great demand. We are concerned with the wellposedness, especially uniqueness and stability, for solutions of the Cauchy problem 1.1 and 1.2. The wellposedness issue is relatively well understood if one removes the diffusion term Au u, thereby obtaining a scalar hyperbolic conservation law; see Kruzhkov [2], LionsPerthameTadmor [21, 22], and Perthame [24, 25]. It is equally well understood if one removes the convection term fu; see [4, 18] and the references cited therein. For the isotropic diffusion, a ij u =,i j, some stability results for entropy solutions have been obtained for BV solutions by VolpertHudjaev [27] in Only in 1999, Carrillo [7] could extend this result to L solutions also see Eymard et al [16], Karlsen Risebro [19] for further extensions, and ChenDiBenedetto [9] handled the case of unbounded entropy solutions which may grow when x is large. Also see Gilding [18] for a theory for isotropic degenerate parabolic equations with isolated degenerate points. In this paper, we establish a wellposedness theory for L 1 solutions of the Cauchy problem 1.1 and 1.2 for general degenerate parabolichyperbolic equations of secondorder, especially including the nonisotropic diffusion case. This relies on two new ingredients. Firstly, the extension from the isotropic to the nonisotropic is not a purely technical issue and we introduce a fundamental and natural chainrule type property, which does not appear in the isotropic case 2
3 and which turns out to be the cornerstone for the uniqueness in the nonisotropic case. Secondly, we extend a notion of kinetic solutions, a new concept in this context, and a corresponding kinetic formulation. The notion of kinetic solutions applies to more general situations than that of entropy solutions as considered in [7, 19] and [9]. The advantage of the new notion is that the kinetic equation in the kinetic formulation is well defined even when the macroscopic fluxes are not locally integrable so that L 1 is a natural space on which the kinetic solutions are posed. Based on this notion, the corresponding kinetic formulation and the uniqueness proof in the purely hyperbolic case introduced in [24], we develop a new, simpler, more effective approach, in comparison with the previous proofs in [7, 19] and [9], to prove the contraction property of kinetic solutions in L 1, especially including entropy solutions. This leads to a wellposedness theory for kinetic solutions in L 1 of the Cauchy problem of 1.1 and 1.2. The main theorems of this paper are the following. Theorem 1.1. Assume that 1.3 and 1.4 hold. Then i For any kinetic solution u L [, ; L 1 with initial data u x, we have ut u L 1, t. ii If u, v L [, ; L 1 are kinetic solutions to 1.1 and 1.2 with initial data u x and v x, respectively, then 1.5 ut vt L 1 u v L 1. iii Furthermore, if u L [,, this kinetic solution is an entropy solution. Theorem 1.2. Assume that 1.3 and 1.4 hold. For u L 1, there exists a unique kinetic solution u C [, ; L 1 for the Cauchy problem 1.1 and 1.2. Ifu L L 1, then the kinetic solution is the unique entropy solution and ut, x u L. In Section 2, we derive a kinetic formulation in a precise manner and describe the notions of entropy solutions and kinetic solutions of the Cauchy problem 1.1 and 1.2. The new ingredient of this formulation is the precise identification of the kinetic defect measure and the degenerate parabolic defect measure, even in the region where ut, x is discontinuous and is only in L 1. To make the points more clearly, in Section 3, we present our new approach by a formal proof to show the contraction property of kinetic solutions. Then Sections 4 6 are devoted to the rigorous proof of the stability of kinetic solutions. In Section 7, we prove the existence of kinetic solutions and entropy solutions of the Cauchy problem 1.1 and
4 In this paper we focus on the prototypical case 1.1. The results and techniques straightforward extends to more general degenerate parabolichyperbolic equations of second order, by combining with the Gronwall inequality, such as 1.6 t u + fu, t, x Au, t, x u =cu, t, x, x, t, where Au, t, x =a ij u, t, x with a ij u, t, x =a ji u, t, x, fu, t, x, and cu, t, x are sufficiently smooth functions, and a ij u, t, xξ i ξ j, for t, x + and u IR. i,j 2. Entropy Solutions, Kinetic Solutions, and Kinetic Formulation Equation 1.1 satisfies a socalled entropy property. To motivate it, we consider a nondegenerate parabolic equation 1.1 in which the matrix Au =a ij u is replaced by Au+εI, and we denote u ε t, x its C 2 solution. Then, for any function S C 2 IR, multiplying the equation 1.1 by S u ɛ yields 2.1 t Su ε + xi ηi S u ε xi S u ε a ij u ε xj u ε ɛ Su ε = m S ε t, x n S ε t, x, where the entropy flux ηi S u is defined up to an additive constant by 2.2 η S i u =a i us u, the entropy dissipation measure m S ε t, x is defined by m S ε t, x :=εs u ε u ε 2, and the parabolic dissipation measure n S ε t, x is given by n S ε t, x :=S u ε a ij u ε xi u ε xj u ε = S u ε σ ik u ε xi u ε 2. In order to understand more about the dissipation measures, we introduce the notations β ik u and β ψ ik u for ψ C IR with ψ : 2.3 β ik u =σ iku, β ψ ik u = ψuσ ik u. 4
5 Then we end up with the two equivalent definitions: 2.4 n ψ ε t, x := 2 K 2. xi β ψ ik uε = ψu ε xi β ik u ε The heart of our investigations is to notice that this equality still holds in the limit ε. It is useful at this stage to derive a priori bounds from the above calculations. After the spacetime integration against S with S convex and S = S =, we obtain K 2 m S ε t, x+n S ε t, x dt dx = S u ε xi β ik u ε + ε u ε 2 dt dx 2.5 Su L 1 S L IR u L 1. The following convenient notations are deduced by the duality C IR; M 1 IR, which replace the exponent S or ψ: Namely, m ψ ε t, x = ψξ m ε t, x, ξ dξ, n ψ ε t, x = ψξ n ε t, x, ξ dξ, with IR 2.6 m ε t, x, ξ =δξ u ε ε u ε 2, n ε t, x, ξ =δξ u ε IR 2, xi β ik u ε where δξ is the Dirac mass concentrated at ξ =. Then we can choose, as a limiting case for smoothness of the entropy Su, the function Su = u ξ + for the parameter ξ, or Su =u ξ for ξ, in 2.5, and we end up with m ε + n ε t, x, ξ dt dx µξ L IR bounded functions that vanish at infinity 2.7 µξ :=1I {ξ>} u ξ + L 1 +1I {ξ<} u ξ L 1. Choosing Su =u 2 /2, we also deduce from 2.5 K 2 m ε + n ε t, x, ξ dt dx dξ = xi β ik u ε + ε u ε 2 dt dx u L 2. As ε, passing to the limit with the above bounds and under the property that u ε t, x converges strongly see 7 below, we end up with the definition of entropy solutions. 5
6 Definition 2.1. An entropy solution is a function ut, x L [, such that i xi β ik u L [, 2, for any k {1,,K}; ii for any function ψ C IR with ψu and any k {1,,K}, 2.9 xi β ψ ik u = ψu xi β ik u L 2 [,, and 2 K 2, 2.1 n ψ t, x :=ψut, x xi β ik ut, x = xi β ψ ik ut, x a.e. iii for any smooth function Su, there exists an entropy dissipation measure m S t, x satisfying that 2.11 m S t, x = IR S ξ mt, x, ξ dξ, with mt, x, ξ a nonnegative measure, such that 2.12 t Su+ xi ηi S u xi aij u xj Su = m S + n S, in D IR + with initial data Sut = = Su. Remark 2.1. Arguing as in 2.7, an entropy solution satisfies 2.13 m + nt, x, ξ dt dx µξ L IR. Remark 2.2. The nonnegative parabolic defect measure nt, x, ξ for an entropy solution ut, x in Definition 2.1 is very simple and given by the following formula: 2, nt, x, ξ =δξ ut, x xi β ik ut, x in the usual sense. Also, the choice Su =u 2 /2gives the L 2 integrability of k K, and yields another useful estimate, as in 2.8: 2.14 m + nt, x, ξ dt dx dξ 1 2 u 2 L 2, provided u L 2. 6 xi β ik u, 1
7 Remark 2.3. When we refer to distributional solutions here, we always mean that the initial data are included in the definition of solutions in the sense of distributions, when a test function does not vanish at t =. That is, a distributional solution ut, x satisfying 2.12 means that, for any test function ϕ D[,, S u t ϕ + η S u x ϕ αij S u x 2 i x j ϕ dt dx = m S + n S ϕdtdx S u x ϕ,x dx, with αij S u = S ua ij u. However, thanks to the chain rule which is postulated in the definition of entropy solutions, several possible variants for the secondorder term are equivalent. Remark 2.4. The main ingredient in Definition 2.1 is the equality in 2.9, which is not always true for a function ut, x. Indeed, if β ik u is discontinuous, this chain rule does not make sense even for any single term in the sums of 2.9. It is natural to assume the equality here because it keeps true in the limiting process u ε t, x ut, x strongly see 7. In the case of a diagonal matrix a ij =for i j, this equality in 2.9 is always true and needs not be included in Definition 2.1. We refer to the appendix for a proof. Therefore, our theory also recovers the results of Carrillo [7] and the extensiosn of [19], [16] and ChenDiBenedetto [9] when the initial data are in L 1 L. On the other hand, we may factor out an S u in the equation 2.12 and obtain a more precise kinetic formulation of nonlinear degenerate parabolichyperbolic equations of secondorder with form 1.1. The new ingredient of this formulation is the identification of the kinetic defect measure mt, x, ξ and the degenerate parabolic defect measure nt, x, ξ in a precise manner, even in the region where ut, x is discontinuous and only in L 1. Compare with the classical kinetic formulation for multidimensional hyperbolic conservation laws by LionsPerthameTadmor in [21, 22]. We introduce the kinetic function χ on IR 2 : +1 for <ξ<u, 2.15 χξ; u = 1 for u<ξ<, otherwise. We notice that, if u L [, ; L 1, then χξ; u L [, ; L 1. 7
8 The simple representation Su = IR S ξ χξ ; u dξ leads to the following kinetic equation, which is equivalent to 2.12: 2.16 t χξ; u+aξ x χξ;u in D IR + with initial data a ij ξ x 2 i x j χξ; u = ξ m+nt, x, ξ χξ; u t= = χξ; u. We are now ready to define the kinetic solutions. Definition 2.2. A kinetic solution is a function ut, x L [, ; L 1 such that i for any nonnegative ψ DIR and k {1,,K}, 2.17 xi β ψ ik u L2 [, ; ii for any two nonnegative functions ψ 1,ψ 2 DIR, 2.18 ψ1 ut, x xi β ψ 2 ik ut, x = xi β ψ 1ψ 2 ik ut, x, a.e.; iii Equation 2.16 holds in D, for some nonnegative measures mt, x, ξ and nt, x, ξ, where nt, x, ξ is defined by 2.19 IR ψξ nt, x, ξ dξ = 2, xi β ψ ik ut, x for any ψ DIR with ψ ; iv the following inequality is satisfied: 2.2 m + nt, x, ξ dt dx µξ L IR. This notion of kinetic solutions applies to more general situations than that of entropy solutions. The advantage is that the kinetic equation is well defined even though the macroscopic fluxes η S u are not locally integrable so that L 1 is a natural space on which kinetic solutions are posed. In the purely hyperbolic case, a full L 1 theory has been developed in Perthame [25]. This approach also covers the socalled renormalized solutions used in the context of hyperbolic scalar conservation laws by Bénilan, Carrillo, and Wittbold [1]. 8
9 Remark 2.5. Any entropy solution is a kinetic solution. Our uniqueness result implies that any kinetic solution in L must be an entropy solution. Therefore, the two notions are equivalent for solutions in L, although the notion of kinetic solutions is more general. Remark 2.6. The degenerate parabolic defect measure nt, x, ξ is no longer defined by the simple formula in Remark 2.2 since d xi β ik u does not belong to L 2 [, in general because 2.14 does not apply. In fact, the only a priori bound used here is that of 2.13 which is also expressed in Remark 2.1 as a corollary of Definition 2.1. The explicit expression in terms of u x for µξ in 2.7 is not fundamental, and the useful information is that µξ is bounded and vanishes at infinity. 3. Contraction Proof: Formal In this section, we give a formal proof for the contraction property of kinetic solutions, i.e. part ii of Theorem 1.1, which takes the advantage of the precise kinetic formulation We will make the proof rigorous in Sections 4 6. Consider two solutions ut, x and vt, x. Denote by pt, x, ξ the kinetic defect measure and by 3.1 qt, x, ξ :=δξ vt, x 2, xi β ik vt, x the parabolic defect measure, which are associated with vt, x. Then our proof consists in using the following microscopic contraction functional introduced in [24, 25]: 3.2 Qt, x, ξ = χξ;ut, x + χξ; vt, x 2χξ;ut, x χξ; vt, x. It is useful for deriving a contraction principle since Qt, x, ξ dξ = ut, x vt, x. The point is to justify the following identities. Firstly, t χξ; ut, x +aξ x χξ;ut, x x 2 i x j a ij ξ χξ; ut, x = sgnξ ξ m + nt, x, ξ, which yields d χξ; ut, x dx dξ = 2 m + nt, x, dx. dt A similar identity holds for vt, x. 9
10 Secondly, we compute d χξ; ut, x χξ; vt, x dx dξ +2 a ij ξ xi χξ; ut, x xj χξ; vt, x dx dξ dt = m + nt, x, ξδξ vt, x δξ+p+qt, x, ξδξ ut, x δξ dx dξ. Then, we have d Qt, x, ξ dx dξ dt =4 a ij ξ xi χξ; ut, x xj χξ; vt, x dx dξ m + nt, x, ξδξ vt, x+p+qt, x, ξδξ ut, x a ij ξ xi ut, x xj vt, x δξ ut, x δξ vt, x dx dξ nt, x, ξδξ vt, x + qt, x, ξδξ ut, x dx dξ, since mt, x, ξ and pt, x, ξ are nonnegative. It remains to notice that, using Remark 2.2 and 3.1, we still have, very formally, nt, x, ξδξ vt, x + qt, x, ξδξ ut, x dx dξ = dx dξ 2 2 δξ ut, x δξ vt, x xi β ik ut, x + xi β ik vt, x dx dξ 2 δξ ut, x δξ vt, x xi β ik ut, x xj β jk vt, x dx dξ =2 =2 δξ ut, x δξ vt, x σ ik ut, x σ jk vt, x xi ut, x xj vt, x dx dξ a ij ξ xi ut, x xj ut, x δξ ut, x δξ vt, x dx dξ. Therefore, we end up with d Qt, x, ξ dx dξ, dt which implies that ut vt L 1 is nonincreasing. This concludes the contraction property j=1
11 4. Contraction Proof: Rigorous To make the proof rigorous, the above argument requires to regularize the linear kinetic equation 2.16 by convolution; This is the first step, in which all notations are also introduced. Then we analyze separately the different terms in the microscopic contraction functional, which requires several steps. Step 1. Regularization. We set ε =ε 1,ε 2, ε 1 for the forward time regularization and ε 2 for the space regularization, and we define ϕ ε t, x = 1 ϕ 1 t 1 ε 1 ε 1 ε d ϕ 2 x, 2 ε 2 where ϕ j,j =1,2,denote the normalized regularizing kernels with ϕ j =1,suppϕ 1 1, in order to allow the time regularization. Next, we use the notations χ := χt, x, ξ =χξ;ut, x, χ ε := χ ε t, x, ξ =χ t,x ϕ ε t, x, ξ, χ := χξ; t, x =χξ;vt, x, χ ε := χ ε t, x, ξ = χ t,x ϕ ε t, x, ξ, and, similarly, m ε := m t,x ϕ ε, p ε := p t,x ϕ ε, n ε := n t,x ϕ ε, q ε := q t,x ϕ ε, where t,x denotes the convolution in time and space. We also need a further regularization in ξ with smoothing kernel ψ δ ξ = 1 δ ψξ δ and use the notation χ ε,δ := χ ε ψ δ. Finally, we need a ξtruncation K R ξ, which is a smooth nonnegative function with bounded support. That is, K R ξ =Kξ/R 1, as R, with Kξ 1 for ξ,, Kξ = 1 for ξ 1/2, Kξ = for ξ 1. The destiny of these parameters is that δ first, R second, and ε finally. The results we will show indicate that the contraction property holds even for any fixed parameter ε, which is an interesting phenomenon. Indeed, for any regularized microscopic contraction functional: 4.1 Q ε t, x, ξ = χ ε + χ ε 2χ ε χ ε, 11
12 we have Proposition 4.1. Under the assumptions of Theorem 1.1, d Q ε t, x, ξ dx dξ. dt Notice that, by convolution, we obtain 4.2 t χ ε + aξ x χ ε 4.3 t χ ε + aξ x χ ε x 2 i x j a ij ξχ ε = ξ m ε +n ε t, x, ξ, x 2 i x j a ij ξ χ ε = ξ p ε +q ε t, x, ξ, in D,, where the initial data are inessential at this stage. Step 2. First terms of the contraction functional. From the space and time regularity of χ ε, we deduce that m ε + n ε t, x, ξ is locally Lipschitz continuous in ξ. Furthermore, multiplying 4.2 by sgnξ, we find d 4.4 χ ε ξ; t, x dx dξ = 2 m ε + n ε t, x, dx. dt Similarly, d 4.5 χ ε ξ; t, x dx dξ = 2 p ε + q ε t, x, dx. dt Step 3. Quadratic term of the contraction functional. Analyzing the quadratic term requires a further ξregularization. We write with t χ ε,δ + aξ x χ ε,δ 2 x i x j a ij ξχ ε ψ δ = ξ m ε +n ε ψ δ +R u 1, R u = R u 1t, x, ξ := div x aξ χ ε ψ δ aξχ ε ψ δ. A similar equation holds for χ ɛ,δ. Therefore, a direct combination gives t χ ε,δ χ ε,δ K R ξ + K R ξ aξ x χ ε,δ χ ε,δ K R ξ χ ε,δ x 2 i x j a ij ξχ ε ψ δ K R ξ χ ε,δ = χ ε,δ K R ξ ξ m ε + n ε ψ δ + χ ε,δ K R ξ R u 1 +χ ε,δ K R ξ ξ p ε + q ε ψ δ + χ ε,δ K R ξ R v x i x j a ij ξ χ ε ψ δ
13 After integration, we obtain 4.6 d χ ε,δ χ ε,δ K R ξ dx dξ = dt 3 R l t+ l=1 7 R l t+r l t, where R l are defined as follows: 4.7 R 1 t = K R ξ χ ε,δ R1 u + χ ε,δ R1 v dx dξ, 4.8 R 2 t = 2 K R ξ l=4 xi σ ik χ ε ψ δ xj σ jk χ ε ψ δ dx dξ, 4.9 R 3 t = K R ξψ δ ξ m ε + n ε ψ δ +p ε +q ε ψ δ dx dξ, R 4 t = K Rξ xi χ ε,δ xj a ij ξχ ε ψ δ + xj σ jk ξχ ε,δ ψ δ xi σ ik ξ χ ε ψ δ dx dξ. The term R 5 t comes from integration by parts in ξ and the following equality: ξ χ ε,δ = ψ δ ξ δξ u t,x,ξ ϕ ε ψ δ, that is, taking into account R 3 t, 4.1 R 5 t = K Rξ χ ε,δ m ε + n ε ψ δ dx dξ. Also, 4.11 R 6 t = K R ξ δξ v t,x,ξ ϕ ε ψ δ n ε ψ δ dx dξ, 4.12 R 7 t = K R ξ δξ v t,x,ξ ϕ ε ψ δ m ε ψ δ dx dξ. The terms R l t, 4 l 7, denote the symmetric terms of R lt, 4 l 7, respectively, where ut, x is replaced by vt, x. Step 4. Estimates of the error terms. following two lemmas. Our goal is now to estimate these error terms in the 13
14 Lemma 4.1. For any 1 p<, when δ, 4.13 R 1 t, R 4 t, R 4t, L p loc IR +, and 4.14 R 3 t In addition, when δ first and R second, m ε + n ε t, x, dx + p ε + q ε t, x, dx, L p loc IR R 5 t, R 5t, in L p loc IR +. Furthermore, for any ɛ, δ, R, 4.16 R 7 t, R 7t, for any t,. It remains to estimate the remaining terms which are more difficult to handle and contain the actual cancellation which motivates our kinetic formulation with the introduction of the parabolic defect measure nt, x, ξ. Lemma 4.2. For any ɛ, δ, R, 4.17 R 2 t+r 6 t+r 6t, for any t,. Lemma 4.1 is proved in 6.1. Lemma 4.2 is proved in 6.2 for entropy solutions the easier case and in 6.3 for kinetic solutions. Step 5. Contraction property. With Lemmas 4.1 and 4.2, we can now conclude the contraction proof. As a consequence of the above lemmas, we can pass to the limit as δ first and R second in 4.4, 4.5, and 4.6. Adding 4.4, 4.5, and substracting twice 4.6 yields d χ ε + χ ε 2 χ ε χ ε dx dξ. dt This complete the proof of Proposition 4.1. Since, when ε, ut vt dx = we conclude the contraction property, that is, χξ; ut, x + χξ; vt, x 2χξ;ut, x χξ; vt, x dx dξ, ut vt dx 14
15 is nonincreasing in t>. The full result ii of Theorem 1.1 is therefore proved after using i which is proved below. 5. Continuity at t =and L 1 Stability In order to obtain the L 1 stability 1.5, it remains to prove the initial time continuity statement i in Theorem 1.1. The proof, first due to [15] in the hyperbolic case, is based on the use of the initial data for all entropies in the weak form of equation At the kinetic level, it amounts to say that χξ; ut, x achieves the initial data χξ; u in the weak form of equation 2.16 see Remark 2.3. We prove only for kinetic solutions which contains the case of entropy solutions and turns out to be rather simple. Proposition 5.1. Let u L 1 and ut, x be a kinetic solution see Definition 2.2. Then, when t +, ut, x u x dx, and t L m + ns, x, ξdsdxdξ, for any L>. L Proof. We follow the proof in [25] and thus we skip some technical details. Using the Dunford Petti theorem, we first consider a nonnegative, strictly convex function Φu with superlinear growth for large u such that Φ = and Φu xdx < +. Definition 2.2 for kinetic solutions yields the following identity: t 5.1 Φut, xdx + Φ ξm + ns, x, ξdsdxdξ = Φu xdx <. This identity can be achieved by first choosing admissible test functions ϕ n t, xφ ξkξ/r with ϕ n t, x 1 { t T,x }, for any T,, and a smooth function Kξ such that Kξ =1for ξ <1, Kξ =for ξ 2, and sgnξkξ, and then taking n first and R second. From 5.1 and the nonnegative sign of m, n, and Φ, we know that ut, x is relatively weakly compact in L 1 loc IRd. Hence we may extract subsequences such that t n and χξ; ut n,x χx, ξ L 1 in L, ut n,x χx, ξdξ in L 1 loc IRd, tn IR m + ns, x, ξds mx, ξ in M 1, 15
16 and sgnξ χx, ξ = χx, ξ 1, mx, ξ, mx, ξdx µξ L IR, where the half arrows and denote the weak convergence and the weakstar convergence in the respective spaces, respectively, and µξ is the function defined in 2.13 and 2.2. Secondly, using the definition of distributional solutions yields that, for any test function φx, ξ in D, IRd+1 φx, ξ χξ; ut,xdxdξ T aξ x φx, ξ a ij ξ xi x j φx, ξ χξ; ut, xdtdxdξ T = φ ξ x, ξm + nt, x, ξdtdxdξ + φx, ξχξ; u xdxdξ. Passing to the limit as T = t n, we deduce that φx, ξ χx, ξdxdξ = φ ξ x, ξ mx, ξdxdξ + φx, ξχξ; u xdxdξ, which implies that the identity holds in the sense of distributions: ξ mx, ξ = χx, ξ χξ; u x. From this, we deduce that the measure mx, ξ is also a function and that, for almost every x, mx, is continuous and vanishes at infinity therefore IR χdξ = u. On the other hand, we know mx, ξ. Therefore, we deduce from Brenier s Lemma [3] that mx, ξ = and χx, ξ =χξ;u x. By the uniqueness of the limit, the whole family χξ; ut, x converges weakstar to χξ; u x as t. Therefore, χξ; ut, x has a trace χ ; u+, = χ ; u on the set t =, defined at least in the weak sense in L 1. On the other hand, we also deduce that, for the locally strictly convex function Φ with superlinear growth at infinity in Step 1, Φut, xdx Φu xdx. The strict convexity of Φ implies that the trace of ut, x on the set t = is in fact defined in the strong sense in L 1 as t : ut, x u x, in L 1, 16
17 and, as a corollary, t Φ ξm + ns, x, ξdsdxdξ. This completes the proof of Proposition Proof of Lemmas 4.1 and 4.2 In this section, we give the proof of Lemmas 4.1 and Proof of Lemma 4.1. Statement 4.16 is a simple consequence of the nonnegativity of the convolution and truncation kernels and the dissipation measures mt, x, ξ and pt, x, ξ. For the term R 1 t, it suffices to consider the representative term: K R ξ χ ε,δ div x aξ χ ε ψ δ aξχ ε ψ δ dx dξ IR d+1 = K R ξ χ ε,δ aξ x χ ε ψ δ aξ x χ ε ψ δ dx dξ IR d+1 K R ξ χ ε,δ t, x, ξ aξ aξ η x χ ε t, x, ξ η ψ δ η dx dξ dη +2 Cϕ χ ε,δ t, x, ξ K R ξ aξ aξ η ψ δ η dη dx dξ. ε 2 IR Notice that the function of ξ defined as K R ξ aξ aξ η ψ δ η dη IR has a uniform bound 2 a L R,R and tends to zero for a.e. ξ, when δ. On the other hand, χ ε,δ t, x, ξ is compact in L p loc, ; L 1 R, R. Therefore, by the Lebesgue Theorem, the expression 6.1 tends to zero in L p loc,, for all 1 p<. This proves the first statement of The fact that the term R 4 tends to zero in L p loc, can be obtained by following the same argument. Indeed, we have R 4 t +3 K Rξ K Cϕ 2 K ε 2 xi χ ε t, x, ξ η xj χ ε t, x, ξ η a ij ξ η σ ik ξ ησ ik ξ η ψ δ ηψ δ η dx dξ dη dη xj χ ε t, x, ξ η IR 2 K Rξ a ij ξ η σ ik ξ ησ jk ξ η ψ δ ηψ δ η dη dη dx dξ, and we conclude as before. 17
18 The results 4.14 and 4.15 concerning R 3 t and R 5 t are much simpler because m ε + n ε t, x, ξ and p ε + q ε t, x, ξ are continuous in ξ, vanish at infinity, and their total masses are dominated by µξ, because of Therefore, we omit the proof Proof of Lemma 4.2: Entropy Solutions. In the case of entropy solutions, the definition of the measure nt, x, ξ allows to write directly the expression R 6 t+r 6 t with explicit convolution terms. This yields R 6 + R 6 = 2 K R ξ ϕ ε t s, x y ϕ ε t s,x y ψ δ ξ us, y ψ δ ξ vs,y yi β ik us, y 2 + yj β jk vs,y 2 ds ds dx dy dy dξ j=1 K R ξ ϕ ε t s, x y ϕ ε t s,x y ψ δ ξ us, y ψ δ ξ vs,y yi β ik us, y On the other hand, we have R 2 t = yj β jk vs,y ds ds dx dy dy dξ. j=1 K R ξ xi ϕ ε t s, x y xj ϕ ε t s,x y ψ δ ξ η ψ δ ξ η σ ik ηχη; us, yσ jk η χη ;vs,y ds ds dx dy dy dξ dη dη. Consider for instance the following term from R 2 t: = = xi ϕ ε t s, x y ψ δ ξ η σ ik η χη; us, y dη dy xi ϕ ε t s, x y β ψ δξ 2 ik us, y dy ϕ ε t s, x y yi β ψ δξ 2 ik us, y dy, where the first equality follows from the definition in 2.3. This term is also equal to the following term from R 6 t+r 6 t: ϕ ε t s, x y ψ δ ξ us, y yi β ik us, y dy, 18
19 as a consequence of the identity between L 2 functions: yi β ψ δξ 2 ik us, y = ψ δ ξ us, y xi β ik us, y, which is exactly statement ii in Definition 2.1 of entropy solutions. Hence, Lemma 4.2 is proved for entropy solutions Proof of Lemma 4.2: Kinetic Solutions. When treating the kinetic solutions, we write again the expression R 6 t+r 6 t, but we only have at hand a weaker form than that in 6.2. That is, R 6 t+r 6t = K R ξ ϕ ε t s, x y ϕ ε t s,x y ψ δ ξ vs,y yi β ψ δξ ik us, y 2 2 =2 +ψ δ ξ us, y j=1 yj β ψ δξ jk vs,y 2 ds ds dx dy dy dξ K R ξ ϕ ε t s, x y ϕ ε t s,x y ψ δ ξ us, y ψ δ ξ vs,y yi β ψ δξ ik us, y j=1 yj β ψ δξ jk vs,y ds ds dx dy dy dξ K R ξ ϕ ε t s, x y ϕ ε t s,x y ψ δ ξ us, y ψ δ ξ vs,y xi β ψ δξ ik us, y xj β ψ δξ vs,y ds ds dx dy dy dξ. jk On the other hand, the expression 6.2 for R 2 t still holds. Therefore, we conclude the proof of Lemma 4.2 if we can justify, for instance, xi ϕ ε t s, x y ψ δ η us, y σ ik ξ η χξ η; us, y dy dη = ϕ ε t s, x y ψ δ ξ us, y xi β ψ δξ ik us, y dy. This is exactly assumptions 2.17 and 2.18 in Definition 2.2. Lemma 4.2 for kinetic solutions. 19 This concludes the proof of
20 7. Existence of Kinetic Solutions We now prove Theorem 1.2 for the existence of kinetic solutions, especially entropy solutions, of the Cauchy problem 1.1 and 1.2. We divide the proof into four steps. Step 1. We first consider the case u W 2,1 H 1 L and we prove the existence of an entropy solution. To do so, we set a ɛ iju := a ij u + ɛi. Then, VolpertHudjaev s theorem [27] implies that, for each ε>, there is a smooth solution u ɛ t, x such that, for t>, we have u ɛ t, L 1 u L 1, uɛ t, L u L, x u ɛ t, L 1 u TV, t u ɛ t, L 1 tu L 1 = fu A ε u u L 1, and inequalities 2.5 and 2.8 hold. Then, for any Ψ with Ψ = ψ D + IR and Ψ = Ψ =, 7.4 2dxdt ψu ε xi β ik u ɛ = ψu ε a ɛ iju ɛ xi u ɛ xj u ɛ dxdt i,j Ψu ε dx C<. By Kolmogorov s compactness theorem and after a standard control of decay at infinity, there is a subsequence still denoted u ɛ t, x such that u ɛ u in C loc [, ; L 1, and thanks to 2.8, xi β ik u ɛ xi β ik u in L 2 [, +. In particular, from this, we obtain property i of Definition xi β ik u L 2 [, +, 2
21 and, 7.6 i 2dxdt 1 xi β ik u 2 u 2 L 2. Furthermore, for any nonnegative ψ Dand k {1,,K}, we have xi β ψ ik uɛ t, x = ψu ɛ t, x xi β ik u ɛ t, x, Notice that, by a strongweak limit, the righthand side converges weakly in L 2 to ψut, x xi β ik ut, x. Also, the lefthand side converges weakly in L 2 to xi β ψ ik ut, x. Therefore, we obtain property ii of Definition 2.1, 7.7 ψut, x xi β ik ut, x = xi β ψ ik ut, x a.e. We may also pass to the limit in equation 2.1. Since, recalling the definition 2.6, nt, x, ξ w lim n ε t, x, ξ, we obtain that ut, x satisfies equation 2.12 in D for some nonnegative measures mt, x, ξ and nt, x, ξ satisfying 2.19 and 2.2 as the argument in 2. Therefore, we have proved that u C [, ; L 1 is an entropy solution and Step 1 is completed. a.e. Step 2. For the general case u L 1, we prove the existence of a kinetic solution directly. The same argument allows to build an entropy solution when u L. Approximate u x byu ɛ W2,1 H 1 L such that u ɛ u, L 1. Then there exists a global entropy solution u ɛ C[, ; L 1 of 1.1 and 1.2 with initial data u ɛ x for each ɛ>. Using the contraction property of Theorem 1.1, u ɛ 1 u ɛ 2 C[, ;L 1 uɛ 1 u ɛ 2 L 1, when ε 1,ε 2. Therefore, {u ɛ } is a Cauchy sequence and there exists u C[, ; L 1 such that u ε t, x ut, x, in C[, ; L 1 when ε, 21
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