Entropy Methods for Reaction-Diffusion Equations: Slowly Growing A-priori Bounds.

Transcription

1 Entropy Methods for Reaction-Diffusion Equations: Slowly Growing A-priori Bounds. Laurent Desvillettes, 1 Klemens Fellner, Abstract In the continuation of [DF], we study reversible reaction-diffusion equations via entropy methods based on the free energy functional for a 1D system of four species. We improve the existing theory by getting 1 almost exponential convergence in L 1 to the steady state via a precise entropy-entropy dissipation estimate, an explicit global L bound via interpolation of a polynomially growing H 1 bound with the almost exponential L 1 convergence, and 3, finally, explicit exponential convergence to the steady state in all Sobolev norms. Key words: Reaction-Diffusion, Entropy Method, Exponential Decay, Slowly Growing A-Priori-Estimates AMS subject classification: 35B4, 35K57 Acknowledgement: This work has been supported by the European IHP network HYKE-HYperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis, Contract Number: HPRN-CT--8. K.F. has also been supported by the Austrian Science Fund FWF project P16174-N5 and by the Wittgenstein Award of P. A. Markowich. 1 CMLA - ENS de Cachan, 61 Av. du Pdt. Wilson, 9435 Cachan Cedex, France Laurent.Desvillettes@cmla.ens-cachan.fr Faculty for Mathematics, University of Vienna, Nordbergstr. 15, 19 Wien, Austria Klemens.Fellner@univie.ac.at 1

2 1 Introduction 1.1 General presentation This paper is part of a general study of the large-time behaviour of diffusive and reversible chemical reactions of the type α 1 A α q A q β 1 A β q A q α i, β i Æ, 1 in a bounded box Ω Ê N N 1. Systems of type 1 are well known in the numerous literature on reactiondiffusion systems. For the large time behaviour of global classical solutions e.g. within invariant domains we refer, for instance, to [Rothe, CHS] and the references therein. For global weak solutions see e.g. [MP, Pie, PS] with references. Many authors e.g. [Zel, Mas, HMP, Web, HY88, HY94, KK] and the references therein have deduced compactness and a-priori bounds from Lyapunov functionals. We recall in particular [Mor, FMS, FHM], where generalized Lyapunov structures of reaction-diffusion systems yield a-priori estimates to establish global existence of solutions. We mention also [Rio, Mul] and the references therein where peculiar Lyapunov functionals are designed to show optimized stability and instability properties for reactiondiffusion systems. As in [DF], we exploit as much as possible the free energy functional of these systems. The basic idea consists in studying the large-time asymptotics of a dissipative PDE by looking for a nonnegative Lyapunov functional Ef and its nonnegative dissipation Df = d Eft along the flow of the dt PDE, which are well-behaved in the following sense: firstly, Ef = f = f for some equilibrium f usually, such a result is true only when all the conserved quantities have been taken into account, and secondly, there exists an entropy/entropy-dissipation estimate of the form Df ΦEf for some nonnegative function Φ such that Φx = x =. If Φ, one usually gets exponential convergence toward f with a rate which can be explicitly estimated. This line of ideas, sometimes called the entropy method, is an alternative to the linearization around the equilibrium and has the advantage of being quite robust. This is due to the fact that it mainly relies on functional inequalities which have no direct link with the original PDE. The entropy method has lately been used in many situations: nonlinear diffusion equations such as fast diffusions [DelPD, CV], equations of fourth order [CCT], Landau equation [DV], etc., integral equations such as the spatially homogeneous Boltzmann equation [TV1, TV, V], or kinetic equations [CCG], [DV1, DV5], [FNS].

3 An approach related to the one we propose shows in a non-constructive way that systems quite more general than 1 have a unique asymptotically stable steady state [Grö, GGH, GH]. In our previous paper [DF], we have proven quantitative exponential convergence to equilibrium with explicit rates all constants are also explicit for the systems modelling the reactions A 1 A and A 1 + A A 3. The proven entropy/entropy-dissipation estimate used global L bounds on the concentrations, which are known for these systems they are consequences of maximum principle type properties. In this paper, we prove exponential convergence in L 1 and consequently for any Sobolev norms for a system with four species A 1 + A 3 A + A 4, for which a global L bound was so far - up to our knowledge - unknown, but for which, at least in 1D, a polynomially growing L bound can be established. We focus on this particular system to present in a simple way the proposed method, which is our primary aim rather than the actual asymptotic result. Note that for the equation that we consider, existence and uniqueness of classical solutions in 1D is a consequence of [Ama] and [Mor, theorem.4]. Global existence of weak solutions in any dimension follows e.g. from [Pie]. The method that we present here is very different from the tools used in these works and can be summarized in this way: Firstly, we prove a polynomially growing L bound for the solution of our equation this a priori bound is therefore called slowly growing. Then, we establish a precise entropy/entropy-dissipation estimate, for which the constant depends logarithmically on the L norm of the solution thanks to a somewhat lengthy, but elementary computation. Thus, a Gronwall type lemma implies almost exponential decay in L 1 towards the steady state. Secondly, we prove an explicit, uniform in time L bound by interpolation of the almost exponential L 1 decay with a polynomially growing H 1 bound. Finally, thanks to this global L bound, the entropy/entropy-dissipation estimate can be used a second time and yields exponential decay towards the steady state. Note that slowly growing a priori bounds have already been used in the context of entropy methods in kinetic theory cf. [TV], as well as interpolation between an explicit decay in weak norm and controlled growth in strong norm cf. [DM]. The last step getting the exponential decay is however a new result in the context of entropy methods. To state the problem, we denote with a i a i t, x, i = 1,, 3, 4, the concentrations of the species A i at time t and point x Ω Ω is a 3

4 bounded interval of Ê, and assume that reactions are taken into account according to the principle of mass action kinetics, which leads to the system t a i d i xx a i = 1 i l a 1 a 3 k a a 4, 3 with the strictly positive reaction rates l, k > and with a i satisfying homogeneous Neumann conditions and the nonnegative initial condition Without loss of generality - we assume x Ω, t, x a i t, x =, 4 x Ω, a i, x = a i, x. 5 l = k = 1, Ω = 1, 6 thanks to the rescaling t 1 k t, x Ω x, a i k l a i. Finally, thanks to a translation, we can suppose that Ω = [, 1]. The solutions of 3 5 conserve the masses, that we assume to be strictly positive: M jk = a j t, x + a k t, x dx = a j, x + a k, x dx >, 7 Ω where we introduce the indices j {1, 3} and k {, 4}. Note that only three of the four M jk s can be chosen independently since they are linked via the total mass M = M 1 + M 34 = M 14 + M 3. 8 Moreover, the conserved quantities provide naturally the following bounds a j t, x dx min {M jk, M jk } := M j, k {,4} sup t sup t Ω Ω Ω a k t, x dx min {M jk, M jk } := M k. 9 j {1,3} When all the diffusivity constants d i are strictly positive, there exists a unique equilibrium state a i, for 3 6 satisfying 7. It is defined by the unique positive constants solving a 1, a 3, = a, a 4, provided a j, +a k, = M jk for j, k {1, 3}, {, 4}, that is: a 1, = M 1M 14 >, a M 3, = M 3 M 1M 3 = M 3M 34 M M >, a, = M 1M 3 >, a M 4, = M 14 M 1M 14 = M 14M 34 M M >. 1 4

5 Finally, we introduce the entropy free energy functional Ea i and the entropy dissipation Da i = d dt Ea i associated to 3 6: Ea i = Da i = 4 Ω a i lna i 1 dx, 11 T d i Ω x ai dxdt + Ω a 1 a 3 a a 4 ln a 1 a 3 a a 4 dx. Outline of the paper: In section, we start by studying a priori bounds entailed by the decay of the entropy functional. These bounds allow to bootstrap an explicit, polynomially-growing in time L bound on the concentrations a i proposition.1, implying global existence of classical solutions this result of existence can be proven by other means, Cf. for example [Ama, Mor]. In section 3, we establish an entropy/entropy-dissipation estimate with a constant depending logarithmically on the polynomially growing L bound proposition 3.1. Hence, by a Gronwall lemma, we obtain in section 4 proposition 4. an almost exponential decay in L 1 towards the steady state a i, of the form M 1 i a i t, a i, L 1 [,1] Ea i, Ea i, e C 1 t lne+t, 1 with a constant C 1 which can be computed explicitly Cf. appendix 5. Furthermore, the almost exponential decay interpolates with a polynomially growing H 1 bound and we obtain an explicit, uniform in time L bound 13. Finally, in return, exponential decay towards the steady state can be proven, and we obtain our main theorem: Theorem 1.1 Let Ω be the interval [, 1], and let d i > for i = 1,, 3, 4 be strictly positive diffusion rates. Let the initial data a i, be nonnegative functions of L Ω with strictly positive masses M jk defined by 7 for j, k {1, 3}, {, 4}. Then, the unique classical solution a i of 3 6 is globally bounded in L : a i t L Ω C,i, 13 and decay exponentially towards the steady state a i, given in 9 1 : M 1 i a i t, a i, L 1 Ω Ea i, Ea i, e C 3 t, 5

6 where C,i and C 3 can be computed explicitly Cf. appendix 5. Remark 1.1 Note that exponential decay towards equilibrium in all Sobolev norms follows subsequently by interpolation of the decay of theorem 1.1 with polynomially growing H k bounds, which follow iteratively for k > 1 from 13 and 68 inserted into the Fourier-representation used in lemma.3 and presented in appendix 5 Sobolev norms of any order are created even if they do not initially exist, thanks to the smoothing properties of the heat kernel. Notations: The letters C, C 1, C,i,... denote various positive constants most of them are made explicit in appendix 5. It will also be convenient to introduce capital letters as a short notation for square roots of lower case concentrations and overlines for spatial averaging remember that Ω = 1 A i = a i, A i, = a i,, A i = A i dx, i = 1,, 3, 4. Finally, we denote f = Ω f dx for a given function f : Ω Ê. A-priori estimates In this section, we establish a polynomially growing L estimate proposition.1 for the solution of eq We start with the Lemma.1 A-priori estimates due to the decay of the entropy Let a i, i = 1,, 3, 4, be solutions of the system 3-6 with initial data such that a i, lna i, L 1 [, 1]. Then, for all T > and with M i defined in 9, x A i L [,T] [,1] 1 1 4d i sup a i lna i L 1 [,1] t [,T] 1 Ω a j, lna j, dx + 4e 1 := C 4,i,14 j=1 a j, lna j, dx + 5e 1 := C 5, 15 i=j sup A i L [,1] M i. 16 t [,T] Proof of lemma.1: Integration of the entropy dissipation 11 yields Ω a i lna i dxt + 4 T d i Ω 6 x A i dxdt Ω a i, lna i, dx,

7 so that since a i lna i e 1, estimates 14 and 15 hold. Then, estimate 16 is just the conservation of masses. Lemma. A-priori bounds in L [, T] [, 1] For i = 1,, 3, 4, the solutions a i of 3-6 with initial data a i, lna i, L 1 [, 1] satisfy for T >, a i L [,T] [,1] C 6,i 1 + T, 17 where the constants C 6,i are stated explicitly in appendix 5. Proof of lemma.: Note first that 1 1 x 1 A i t, x A i t, y dy = u A i t, u dudy Hence A i t, x 1 u=y u A i t, u du + 1 u= A i t, u du, u A i t, u du. which yields 17 using 16 and 14, thanks to the following computation: a i L [,T] [,1] T sup A i t, y 1 A it, x dx dt y [,1] T 1 M i ua i t, u dudt + Mi T C 6,i1 + T. The next technical lemma provides classical polynomially growing bounds for the solution of the 1D heat equation, which can be proven in an elementary way. The main steps of this proof are explained in appendix 5, together with a formula for the constant C 7 which appears in 19. Lemma.3 Explicit L r bounds r 1 for the 1D heat equation Let a denote the solution of the 1D heat equation t >, x [, 1], with constant diffusivity d a with homogeneous Neumann boundary condition, i.e. t a d a xx a = g, x at, = x at, 1 =, 18 and assume for the initial data a, x = a x and for the source term gt, x that a L [, 1], g L p [, + [, 1]. Then, for the exponents r, p 1 and q [1, 3 satisfying 1 r + 1 = 1 p + 1 q and for all T >, the norm a L r [,T] [,1] grows at most polynomially in T like a L r [,T] [,1] T 1/r a L [,1] + C T 1 q +1 g L p [,T] [,1]. 19 7

8 Next, we apply lemma.3 to the right-hand side g = a 1 a 3 a a 4 of our system, which is bounded in L 1 by lemma.. As result, we obtain an L r bound with r < 3 on the a i and thus an improved bound on g. Hence, after three iterations detailed below, we obtain that the L norm increases at most polynomially in time: Proposition.1 Let a i, i = 1,, 3, 4, be solutions of the system 3-6 with bounded initial data a i, L Ω. Then, for T >, a i L [,T] [,1] C 8,i 1 + T 1. The constants C 8,i and the constants in the proof are stated in appendix 5. Proof of proposition.1: By lemma., we have a 1 a 3 a a 4 L 1 [,T] [,1] 1 C 6,i 1 + T. Then, by lemma.3 with p = 1 and r = q [1, 3, for i = 1,, 3, 4, a i L r [,T] [,1] T 1 r a i, L [,1] + C 7 4 C 6,i 1 + T 1 r T, a i, L [,1] + 3 C 7 4 C 6,i1 + T 1 r +3. Next, for any s [, 3, a 1 a 3 a a 4 L s [,T] [,1] C T s +3. Using again lemma.3, but with p = s, q [1, 3 and r [1,, it follows that a i L r [,T] [,1] T 1 r a i, L [,1] + C 7 C T 1 q T s +3 C 16,i 1 + T 1 r +9, 1 since T 1 q +1 T s +3 = T 1 r +9, and with the constants C 16,i given in appendix 5. Then, for s [,, we see that a 1 a 3 a a 4 s L [,T] [,1] 1 a i L s [,T] [,1] C16,i 1 + T s +9, and, secondly, by a last application of lemma.3 with p = s, r =, and 1 = 1 + 1, p q a i L [,T] [,1] a i, L [,1] + C 7 4 C 16,i 1 + T 1 q T s +9 C 8,i 1 + T 1, since T 1 q +1 T s +9 = T 1, and with the constant C 8,i defined in appendix 5. 8

9 3 Entropy/entropy dissipation estimate In this section, we prove proposition 3.1, which details an entropy/entropydissipation estimate for Ea i, Da i defined in 11. The proof uses the technical but elementary lemmata 3.1 and 3.. Despite being lengthy, we believe that the lemmata 3.1 and 3. provide a strategy which extends to more general reaction-diffusion systems. In particular, in the special case of spatial-independent nonnegative concentrations, lemma 3.1 establishes a control of a L -distance towards the steady state in terms of a reaction term, which - due to the conservation laws 7 - can t cease until the steady state is reached. Lemma 3. generalizes this control to spatial-dependent concentrations. We begin with the : Lemma 3.1 Let A i,, i = 1,, 3, 4, denote the positive square roots of the steady state 1. Let A i be constants satisfying the conservation laws 7, i.e. A j + Ak = A j, + A k, for j, k {1, 3}, {, 4}. Then, A i A i, C 9 A 1 A 3 A A 4, 3 where C 9 is given in appendix 5. Proof of lemma 3.1: The proof exploits the ansatz A i = A i, 1 + µ i, 1 µ i, for i = 1,, 3, 4. 4 The conservation laws 7, more precisely the relations A 1, µ 1 + µ 1 + 1i A i, µ i + µ i =, i =, 3, 4, 5 allow to express µ, µ 3, and µ 4 as functions of µ 1 : µ i = µ i µ 1 = ia 1, µ A 1 + µ 1, i =, 3, 4. 6 i, The function µ 1 µ 3 µ 1 is monotone increasing, while µ 1 µ µ 1 and µ 1 µ 4 µ 1 are monotone decreasing. Moreover, µ i µ 1 = if and only if µ 1 = for i =, 3, 4. Since µ µ 1, µ 3 µ 1, µ 4 µ 1 are real, µ 1 is restricted to µ 1,min µ 1 µ 1,max, 7 9

10 with µ 1,min = min{a 1,,A 3, } A 1,, µ 1,max = 1 + Due to the above monotonicity properties, we see that 1 + min{a,,a 4, } 8. A 1, 1 µ 3 µ 1,min µ 3 = µ 3 µ 1,max, 9 1 µ i µ 1,max µ i = µ i µ 1,min, i =, 4. 3 We now quantify how µ 1 µ i µ 1 for i =, 3, 4 are close to proportional to µ 1. In particular, for µ 3, we Taylor-expand 1 + A 1, A 3, µ 1 + µ 1 = ζ A 1, 1+A 1, /A 3, ζ+ζ A3, µ 1, for some ζ, µ 1, and consider the remainder R 3 µ 1 = A 1, 1+ζ = A 3, 1 1+A 1, /A 3, ζ+ζ A 1, µ A 1, A3, µ 1 + µ 1, for µ 1 [µ 1,min, µ 1,max ]. It is straightforward that R 3 µ 1 is continuous at µ 1 = with R 3 = A 1,, and monotone increasing or decreasing in µ 1 [µ 1,min, µ 1,max ] if and only if A 1, < A 3, or A 1, > A 3,, respectively. Therefore, so that < R 3 µ 1,min < R 3 µ 1,max A 3,, for A 1, A 3,, < R 3 µ 1,max < R 3 µ 1,min < A 1,, for A 1, A 3,, < R 3 max{a 1,, A 3, }. 31 For µ and analogously for µ 4, we expand 1 A 1, µ A 1 + µ 1 = 1 1+ζ, for some ζ, µ 1, and consider the remainder R µ 1 = A 1, 1+ζ = 1 A 1, /A, ζ+ζ A, 1 A 1, µ 1 1 A 1, 1 A 1, /A, ζ+ζ A, µ 1, 1 A 1, A, µ 1 + µ 1 which is continuous with R = A 1,, and increases with respect to µ 1. Therefore, < R µ 1,min < R µ 1,max A 1, + A 1, + min{a,, A 4, }, 1,

11 so that finally, < R, R 4 A 1, + A 1, + min{a,, A 4, }. 3 Using the ansatz 4 to expand 3 and using the identity A 1, A 3, = A, A 4,, we see that in order to prove lemma 3.1, we only have to establish that A 1, µ 1 + A, µ + A 3, µ 3 + A 4, µ 4 A 1, A 3, µ 1 + µ 3 + µ 1 µ 3 µ µ 4 µ µ 4 C 9, 33 for µ 1 [µ 1,min, µ 1,max ]. Considering the numerator of 33, we estimate thanks to 31, 3 that A i, µ i µ 1A 1, 1 + R + R 3 + R 4 µ A, A 3, A 1A 1, A 3, C 9, 34 4, where C 9 is given in the appendix 5. Regarding the denominator of 33, we assume first that µ 1 <. Then, thanks to the properties of monotonicity of µ 1 µ i µ 1, we observe in the sum µ 1 + µ 3 + µ 1 µ 3 + µ + µ 4 + µ µ 4 that only the term µ 1 µ 3 is nonnegative and all the other terms are nonpositive. Moreover, we know in this case that 1 µ 1 and 1 µ 3 ; and therefore µ 3 µ 1 µ 3, implying µ 1 + µ 3 + µ 1 µ 3 µ µ 4 µ µ 4 µ 1 µ µ 4 µ µ 4 µ If we secondly consider the case µ 1 >, only the term µ µ 4 is nonpositive and 1 µ as well as 1 µ 4, therefore µ µ µ 4 and µ 1 + µ 3 + µ 1 µ 3 µ µ 4 µ µ 4 µ 1 + µ 3 + µ 1 µ 3 µ 4 µ Altogether, by 35 and 36, we estimate the denominator of 33 by A 1, A 3, µ 1 + µ 3 + µ 1 µ 3 µ µ 4 µ µ 4 A 1, A 3, µ 1, which proves with 34 that we can take the constant 73, and lemma 3.1 is obtained. The following lemma extends lemma 3.1 to nonnegative functions A i which satisfy the conservation laws 7. Lemma 3. Let A i,, i = 1,, 3, 4, denote the positive square roots of the steady state 1, and A i be measurable, nonnegative functions satisfying the 11

12 conservation laws 7, i.e. A j + A k = M jk = A j, + A k, {1, 3}, {, 4}. Then, for j, k A i A i, C 1 A 1 A 3 A A 4 + C 11 A i A i, 37 where { { } { }} 4M 4M C 1 = max C 9, max, max, 38 j=1,3 M j M j4 k=,4 M 1k M 3k with C 9 defined in 73 and C 1 M14 M 3 + M, if A i ε i for some i = 1,, 3, 4, C 11 = { C 9 M14 M max } { } M Ai,,,3,4 ε i + max,,3,4 ε i, else. 39 Here M Mj M j4 ε j = 1 + M j + M j4 1, j = 1, 3, 4 M j + M j4 M M M1k M 3k ε k = 1 + M 1k + M 3k 1, k =, M 1k + M 3k M Proof of lemma 3.: In order to apply lemma 3.1, we expand around the mean values A i = A i + δ i x, δ i =, i = 1,, 3, 4, 4 and consider the ansatz in lemma 3.1: A i = A i, 1 + µ i, 1 µ i, 43 which preserves the relations 5 and thus all the sequel of lemma 3.1. The ansatz 4, 43 implies readily for the right-hand side of 37 that A i A i = A i A i = δ i, 44 Since δ i A i + A i = A i A i, 45 1

13 it follows that 1 A i = A i, 1 + µ i δ A i + A i. 46 i For the left-hand side of 37, we use 46 to expand A i A i, = A i, µ i + A i, δi. 47 A i + A i Thus the expansions in terms of δi is unbounded for vanishing A i A i and we consider firstly the Case A i ε i: leaving the case for small A i for later. We factorize A 1 A 3 A A 4 = A 1 A 3 A A 4 + A 1 A 3 A A 4 δ 1 δ 3 δ δ 4 + A 1 δ 3 + A 3 δ 1 + δ 1 δ 3 A δ 4 A 4 δ δ δ Since A i A i by Jensen s inequality and A 1 A 3 M 14M 3, A A 4 M 14 M 3 by the conservation laws 7, we estimate the second term on the right-hand side of 48 using Young s inequality: A 1 A 3 A A 4 δ 1 δ 3 δ δ 4 A 1 A 3 A A 4 δ 1 + δ + δ 3 + δ 4 M 14 M 3 δ 1 + δ + δ 3 + δ Then, we insert 46 recalling A 1, A 3, = A, A 4, into A 1 A 3 A A 4 = A 1, A 3, µ 1 + µ 3 + µ 1 µ 3 µ µ 4 µ µ 4 A 1 A 3 A A A 3 δ A 1 δ 3 δ1 δ 3 + A 3 δ 1 A 1 +A 1 A 1 +A 1 A 4 δ A +A A 3 +A 3 A δ 4 + A 4 +A 4 δ δ 4 A +A A 4 +A 4 A 1 +A 1 δ δ 4 A +A A 3 +A 3 A 4 +A 4. 5 For the second factor on the right-hand side of 5, we estimate like above A 1 A 3 A A 4 M 14 M 3 and use 45 to compute A 3 δ 1 + A 1 +A 1 A 1 δ 3 A 3 +A 3 δ 1 δ 3 = 1 A 3 +A 3 A 1 +A 1 A 3 +A 3 13 A 1 1δ +A A 1 +A 1 A 3 3δ +A 3

14 and we compute in the same way the product proportional to δδ 4. Thus, by A i A i < M for all i = 1,, 3, 4, we obtain M 14 M 3 A 3 +A 3 δ A 1 +A 1 + A 1 +A 1 δ 1 A 3 +A 3 A 4 +A 4 δ 3 A +A A +A A 4 4δ +A 4 { } M 14 M 3 max M δ1 +,,3,4 δ + δ 3 + δ Therefore, inserting 47 into the left-hand side of 37 and combining 44 and for the right-hand side of 37 we have to prove that + A i A i, µ i C 1A 1, A 3, µ 1 + µ 3 + µ 1 µ 3 µ µ 4 µ µ 4 { } { } C 11 C 1 M14 M max M max A i, 4 δ i A i i A i +A i. i When C 1 C 9 with C 9 stated in 73, then lemma 3.1 see 33 implies 4 A i, µ i C 9A 1, A 3, µ 1 + µ 3 + µ 1 µ 3 µ µ 4 µ µ 4 and we look for C 11 C 9 M14 M max,,3,4 { M A i } { } A + max i,. 5,,3,4 A i We now treat the case Case A i ε i : More precisely, we suppose that A i ε i, where ε i are constants to be specified later. In particular for A 1 A 11/ ε1, we estimate using A 3 < M, A M 1, A 4 M 14 and A A4 = A δ A 4 δ 4 with 7 for the product A A 4 that A 1 A 3 A A 4 = A 1 A3 A1 A 3 A A 4 + A A4 ε 1 M M1 M 14 + M 1 ε 1 M 14 ε 1 A 4 δ A δ 4. Moreover by 7, a straightforward expansion yields 53 A i A i, M. 54 Thus, combining the left-hand side of 37 with 54 and the right-hand side with 48, 49, and 53 where A 4, A M, we must prove that M C 1 M 1 M 14 ε 1 M M1 M 14 ε 1M 1 + M 14 + ε C 11 C 1 M14 M 3 C 1 M δ1 + δ + δ 3 + δ

15 We treat the first bracket on the right-hand side of 55. After neglecting ε 4 1, we denote the nonnegative solution of the in terms of ε quadratic equation xy ε M xy ε x + y = h for h xy by M xy εx, y, h := x + y + M xy x + y + xy h x + y. 56 In the present case, where x = M 1 and y = M 14, choosing in particular h = xy confirms 55 with C 1 M = } 4M h M 1 M 14, C 11 C 1 M14 M 3 + M for A 1 ε 1 := εm 1, M 14, M 1 M 14., 57 Similarly, for the cases A i ε i, i =, 3, 4, we obtain the same C 11 and 4M C 1, for A 3 ε 3 := εm 3, M 34, M 3 M 34, 58 M 3 M 34 4M C 1, for A k M 1k M ε k := εm 1k, M 3k, M 1k M 3k, k =, 4, 3k and this yields 38 and 39. We are now in position to state the entropy/entropy-dissipation estimate for E, D defined in 11, which holds for admissible functions regardless if or if not they are solutions at a given time t of eq Proposition 3.1 Let a i be measurable functions from [, 1] to Ê such that a i a i L [,1], and 1 a j + a k = M jk for j, k {1, 3}, {, 4}. Then, Da i 4 { 1 min, min{d } 1, d, d 3, d 4 } Ea i Ea i, 59 C 1 C 1 C 11 P[, 1] where P[, 1] is the Poincaré constant of interval [, 1], C 1 C 1 M jk is defined in 38, C 11 C 11 M jk in 39, and C 1 a i L [,1], M jk = max i { Φ ai L [,1], a i, }. 6 Here, Φ is the function defined by the formula Φx, y = x lnx lny x y x y, Φx, y = Olnx

16 Proof of proposition 3.1: Using the inequality a 1 a 3 a a 4 lna 1 a 3 lna a 4 4A 1 A 3 A A 4 and Poincaré s inequality, we obtain the estimate Da i 4 A 1 A 3 A A 4 + 4d i Ai A i PΩ. 6 We show in the sequel that the right-hand side of 6 is bounded below by the relative entropy Ea i Ea i,. First, we use the conservation laws 7 to rewrite the relative entropy as Ea i Ea i, = Ω a i ln a i a i a i, dx a i, and we use the boundedness of the function Φ defined in 61, see [DF, lemma.1] to estimate Ea i Ea i, C 1 A i A 1,, 63 with C 1 as defined in 6. The statement of proposition 3.1 follows now from lemma 3. by comparison with 6. 4 Estimates of convergence towards equilibrium In this section, we use the estimates of the two previous sections in order to obtain proposition 4. and theorem 1.1. We begin with a Cziszar-Kullback type inequality relating convergence in entropy with L 1 convergence. Proposition 4.1 For all measurable functions a i : [, 1] Ê +, i = 1,, 3, 4, for which 1 a j+a k = M jk for j, k {1, 3}, {, 4}, we have the inequality Ea i Ea i, M 1 i a i a i, 1, with M i defined in 9 and for the entropy functional Ea i defined in 11. Proof of proposition 4.1: We define qa i = a i ln a i a i and rewrite Ea i Ea i, = Ω a i ln a i a i dx + 16 qa i qa i,. 64

17 Using the conservation laws 7, we define moreover Q jk M jk, a j = qa j + qm jk a j = Q jk M jk, a k for a j, a k [, M jk ], and rewrite the second sum on the right-hand side of 64 as qa i qa i, = Q jk M jk, a j Q jk M jk, a j, +Q j k M j k, a j Q j k M j k, a j, = Q jk M jk, a k Q jk M jk, a k, +Q j k M j k, a k Q j k M j k, a k,, with j j and j, j {1, 3} and k k and k, k {, 4}. Since the derivatives Q jk and Q jk satisfy and Q jk M jk, a j + Q j k M j k, a j = Q jk M jk, a k + Q j k M j k, a k =, Q jk M jk, a j 4 M jk, Q jk M jk, a k 4 M jk, we Taylor-expand 64 where the first order terms vanish due to a 1 a 1, = a 3 a 3, and a a, = a 4 a 4,, respectively and get qa i qa i, M 1 i a i a i,. Secondly, for the first term on the right-hand side of 64, we estimate with the classical Cziszar-Kullback-Pinsker inequality Cf. [Csi] a i ln a i dx 1 a i a i 1 a i a, i Ω for which moreover a i M i. Alltogether, we obtain by Young s inequality a i a i, 1 a i a i 1 + a i a i, Ea i Ea i, a i a i, 1 M i. This ends the proof of proposition 4.1. We now are in a position to state the 17

18 Proposition 4. Let d i > for i = 1,, 3, 4 be strictly positive diffusion rates. Let the initial data a i, be nonnegative functions of L [, 1] with strictly positive masses M jk for j, k {1, 3}, {, 4}. Then, the unique classical solution t, x a i t, x to eq. 3 6 satisfies for M i defined in 9 and E in 11 the decay 1, i.e. M 1 i a i t, a i, L 1 [,1] Ea i, Ea i, e C 1 t lne+t, with a constant C 1 which can be computed explicitly Cf. appendix 5. Proof of proposition 4.: Thanks to the entropy identity d Ea dt i = Da i, proposition 3.1 yields d dt lnea i Ea i, 4 { 1 C 1 t min, min{d } 1, d, d 3, d 4 }, 65 C 1 C 11 P where C 1 t = max,,3,4 {Φ a i L [,t] [,1], a i, } with Φx, y defined in 61 this function is monotone increasing in x, Cf. [DF], lemma.1, and C 1, C 11 and P defined in proposition 3.1. Moreover, it is easy to see that for k > 1, Φky, y = k lnk k 1 k 1 k + 1 k 1 lnk, k > Note that the factor k +1/ k 1 is strictly monotone decreasing in k. Next, we know thanks to lemma.1 that a i L [,t] [,1] C 8,i 1+t 1. Thus, in order to apply 66 with e.g. k, we estimate a i L [,t] [,1] max{c 8,i, a i, } 1 + t 1 so that } a i,, a i, Φ a i L [,t] [,1], a i, Φ +1 1 ln max and therefore C 1 t + 1 max,,3,4 max { C8,i a i,, { C8,i a i,, } + ln 1 + t t 1 { } C8,i ln, ln + ln 1 + t 1. a i, Next, we notice that T dt max{ln C 8,i a i,, ln } + ln1 + t 1 1 T max{ C 8,i a i,, ln} + 1 lne + T, 67 18

19 since both sides vanish at T = and the time-derivatives of the left-hand side can be estimated below by > 1 max{ln C 8,i a i,, ln } + 1 lne + T 1 1 max{ C 8,i a i,, ln } + 1 lne + T 1 1 max{ C 8,i a i,, ln} T 1, lne + T e + T lne + T which is the time-derivative of the right-hand side of 67. Finally, estimate 1 follows from integrating 65 on [, T] and the Cziszar-Kullback type proposition 4.1. We now present the proof of theorem 1.1, which is based on interpolation properties, and a second application of the entropy/entropy-dissipation estimate proposition 3.1. Proof of theorem 1.1: To establish an H 1 bound on the solution of eq. 3 6, proposition.1 and 14 yield T inf x a i L t [,T] [,1] xa i L [,T] [,1] 4C 4,i C 8,i 1 + T 1, for all T >. Since the function T 1 + T 19 assumes its minimum value at time T = /19 /1, there exists a time τ [, /19 /1 ] when x a i τ L [,1] 4C 4,i C 8,i Next, multiplying eq. 3 formally with xx a i yields with Young s inequality d dt 1 1 x a i dx+d i xx a i dx 1 a 1 a 3 a a 4 L 4d [,1] +d i i 1 xx a i dx. We integrate over a time interval T > /19 /1 τ this formula and obtain x a i T L [,1] xa i τ L [,1] + 1 4d i a 1 a 3 a a 4 L [,T] [,1]. Using the bound, i.e. a 1 a 3 a a 4 L [,T] [,1] 4 C 16,i 1 + T 19, we obtain x a i T L Ω < C T 19 for T > /19 /1, 68 19

20 with the constant C 17 given in the appendix 5. This formal argument can be made rigorous by approximations of the solution see e.g. [MP]. Next, we use see e.g. [Tay] the Gagliardo-Nirenberg-Moser interpolation inequality a i L [,1] G[, 1] x a i 1 L [,1] a i 1 L [,1]. 69 Then, interpolating the almost exponentially decaying L 1 norm of proposition 4. for T > /19 /1 τ, we get a i T L [,1] a i, + a i a i, L [,1] a i, + G[, 1] x a i a i, 1 L [,1] a i a i, 1 4 L [,1] a i a i, 1 4 L 1 [,1] C 13,i, where 68, proposition.1 and proposition 4. lead to the constants C 13,i given by 74 in appendix 5. Moreover, for < τ /19 /1, the L bound of theorem 1.1, i.e. the value of C,i 7 follows from proposition.1. Finally, using this global L bound, the right-hand side of 65 is bounded below by a constant and the exponential decay stated in the theorem can be obtained by the standard Gronwall s lemma. 5 Appendix In order to convince the reader that all constants in this work are explictly computable, we provide the following formulas: Lemma 5.1 Explicit constants C 3 = C 7 = d 1 q a C 9 = j 4min 4q 3q qq 1 C 1 = C,i n 4min 1 o, min{d 1,d,d 3,d 4 } C 1 C 11 P[,1] ff,ln + 1 a i, +1 max,,3,4 j C8,i { 1 C 19 8,i, < t C 13,i, t > 1, min,,3,4 {d i } C 1 C 11 P[,1] max {ΦC 13,i,a i, },,3,4 π q 1+max{4 A 1, A,1} 3, A 3, +q qq q ff 19 /1, 19 /1, «,, C 6,i = M i M i + C 4,i, +q q 3 qq 1 1 qq 1 +q + d 1 a 3+ q 1 +q 7 1 q 71, C 8,i = a i, L [,1] + 3 C 7 4 C 16,i, A, + 1 A1, + A 1, +min{a,,a 4, } A 4, A 3, C 13,i = a i, + GΩ C C 1 4 8,i 3 M i Ea i, Ea i, 1 8, C 14, 74

21 { 1 C 14 = sup 1 + t 59 4 t [, exp n o t min 1, min{d 1,d,d 3,d 4 } C 1 C 11 P[,1] lne+t j «C8,i +1 max ff+ 1,,3,4 a i, } C 15 = 4 ai, L [,1] + 3C 4 7 C 6,i, 75 C 16,i = a i, L [,1] + 3 C 7 C 15, 76 C 17 = 4C 4,i C 8,i d i 4 C 16,i, 77 where M i is defined in 9, C 4,i is given in 14, C 1 is defined in 38 depending on C 9 given in 73, C 11 is defined in 39, C 15 in 75, C 16,i in 76, and the function Φ is given in 61. Moreover, P[, 1] denotes the Poincaré constant of [, 1] and G[, 1] denotes the Gagliardo-Nirenberg- Moser constant in 69., We also provide a short proof of lemma.3 for the sake of completeness. Proof of lemma.3: The proof uses Fourier series, which simplify when 18 is mirrored evenly around x =, i.e. when the functions are extended like { at, x x [, 1], ãt, x = 78 at, x x [ 1, ], and when g and ã are defined analogously. Then, the eigenvalue-problem ϕ xx = λ ϕ on [ 1, 1] with homogeneous Neumann boundary and periodicity conditions is satisfied by the eigenvalue-eigenfunction pairs λ k, ϕ k x = kπ, coskπx for k =, 1,,... and yields the Fourier representation ãt, x = 1 1 ãy dy + + t 1 gs, y dyds + 1 k=1 e λ kd at 1 k=1 t eλ kd at s ãy ϕ 1 k y dy ϕ k x 1 gs, y ϕ 1 kydy Thanks to Poisson s summation formula, we can write down ãt, x = 1 1 ãy π π t 1 gs, y 1 k= k= ds ϕ k x.79 1 e k+x y 4dat dat dy 8 1 e k+x y 4dat s dyds. dat s 1

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