Regularity Analysis for Systems of Reaction-Diffusion Equations

Transcription

1 Regularity Analysis for Systems of Reaction-Diffusion Equations Thierry Goudon 1, Alexis Vasseur 2 1 Team SIMPAF INRIA Futurs & Labo. Paul Painlevé UMR 8524, C.N.R.S. Université Sciences et Technologies de Lille, Cité Scientifique, F Villeneuve d Ascq cedex, France Thierry.Goudon@math.univ-lille1.fr 2 Department of Mathematics University of Texas at Austin Austin, Texas 78712, USA vasseur@math.utexas.edu Abstract This paper is devoted to the study of the regularity of solutions to some systems of Reaction Diffusion equations. In particular, we show the global boundedness and regularity of the solutions in 1D and 2D, and we discuss the Hausdorff dimension of the set of singularities in higher dimensions. Our approach is inspired by De Giorgi s method for elliptic regularity with rough coefficients. The proof uses the specific structure of the system to be considered and is not a mere adaptation of scalar techniques; in particular the natural entropy of the system plays a crucial role in the analysis. Key words. Reaction-Diffusion Systems. Regularity of solutions. AMS Subject classification. 35Q99, 35B25, 82C7 1 Introduction This paper is devoted to the analysis of the following system of Reaction-Diffusion equations t a i x D i x a i ) = Q i a), i {1,..., p}, p p ) Q i a) = µ i ν i ) k f a ν j j k b a µ j j, 1.1) j=1 j=1 a i t= = a i. This work has been done while Th. G. was visiting ICES and the University of Texas at Austin. Thanks are addressed for their warm hospitality and port. We also thank J.-F. Coulombel who read preliminary versions and has suggested many improvements. A. Vasseur was partially ported by a NSF grant. 1

2 The equation holds for t and the space variable x lies in where - either = R N, - or R N is a bounded domain with smooth boundary and the system is completed by imposing the Neumann boundary condition D i x a i νx) =, where νx) stands for the outer normal vector at x. The matrices D i x) are required to satisfy { Di L ) ) N N, D i x)ξ ξ α ξ 2, α > for any ξ R N, x. 1.2) Such a system is intended to describe e.g. the evolution of a chemical solution: the unknown a i stands for the density of the species labelled by i {1,...,p} within the solution. Accordingly, the unknowns are naturally non negative quantities: a i. The right hand side of 1.1) follows from the mass action principle applied to the reversible reaction ν i A i µ i A i, where the µ i and ν i s the so-called stoichiometric coefficients are integers. The positive) coefficients k f and k b are the rates corresponding to the forward and backward reactions, respectively. The key feature relies on the conservation property There exists m 1,...,m p ) N p, m i, such that 1.3) m i µ i = m i ν i which implies d dt m i a i dx =, while the entropy is dissipated: we set K = k b /k f, then p Q i a) lna i /K 1/pµ i ν i )) ) = k f a µ i i K p ) a ν i i ln p p K a µ i i a ν i i. 1.4) Actually, we realize readily that, up to a change of unknowns, we can reduce the situation to m i = 1, k f = 1 = k b. 2

3 From now on, we adopt this framework and a crucial role will be played by the quantity µ = µ i = ν i, where the coefficients µ i and ν i are still integers. As we shall see below, our analysis will be faced to restriction on the space dimension N and the parameter µ; one of the most interesting situation we are able to deal with is the following example corresponding to 4 species subject to the reactions which leads to A 1 + A 3 A 2 + A 4, Q i a) = 1) i+1 a 2 a 4 a 1 a 3 ), 1.5) We refer for a thorough introduction to the modeling issues and mathematical properties of such reaction diffusion systems to [12, 14, 15, 2, 21, 22, 24, 28, 31]. Information can also be found in the survey [7] with connection to coagulation-fragmentation models and in [25] for applications in biology. Let us also mention that 1.1) can be derived through hydrodynamic scaling from kinetic models, see [3]. In this contribution we are interested in the derivation of new L estimates and we investigate the regularity of the solutions of 1.1). Quite surprisingly, the question becomes trivial when the diffusion coefficients vanish: for D i =, we are dealing with a system of ODEs and the conservation property 1.3) guarantees that solutions are globally defined. Conversely, certain reaction diffusion systems might exhibit blow up phenomena, see e.g. [23], as it is also well known when considering non linear heat equations [16, 35]. Therefore global well-posedness and discussion of smoothing effects is an issue. While standard techniques can be applied to show the existence of a smooth solution locally in time, with, say, initial data in L 1 L ), extending the result on arbitrarily large time interval is a tough question. Roughly speaking, this is due to a lack of estimates since the only natural bounds are provided by the mass conservation 1.3) and the entropy dissipation 1.4). In turn this leads to an L 1 framework which is quite uncomfortable and with such a bound it is even not clear that the right hand side of 1.1) makes sense. Recently, by using the tricky techniques introduced in [27, 26], it has been shown in [11] that the solutions of 1.1) in the quadratic case 1.5) are a priori bounded in L 2, T) ) so that the nonlinear reaction term makes sense at least in L 1. Besides, [11] establishes the global existence of weak solutions of 1.1), 1.5). The dissipation property 1.4) is also the basis for studying the asymptotic trend to equilibrium [9, 1] in the spirit of the entropy/entropy dissipation techniques which are presented e.g. in [34] we refer also to [2] for further investigation of the large time behavior of nonlinear evolution systems using the entropy dissipation). Our approach is inspired by De Giorgi s methods for studying the regularity of solution of diffusion equations without requiring the regularity of the coefficients, see [8]. The crucial step consists in establishing a L estimate on the solution, and, then, we can deduce that hölderian regularity holds. This approach has been used in [33] to obtain an alternative proof to the regularity 3

4 results for the Navier-Stokes equation [5, 18] and it also shares some features with the strategy introduced in [29, 3]. It has also been applied to study convection-diffusion equations [19] and regularity for the Quasi-geostrophic equation [6]. In the framework of Reaction-Diffusion systems, the De Giorgi s method was first introduced in [1]. In our situation, it is worth pointing out that the proof highly relies on the structure of the whole system and the argument is not a mere refinement of a scalar approach. Indeed, we use in a crucial way the properties of the underlying entropy functional. As we shall see however, restrictions appear between the space dimension N and the degree of nonlinearity of the reaction term measured by means of µ. For this reason, the L estimates can be proved in 2D for the quadratic operator 1.5) or in 1D considering cubic terms. Theorem 1.1 We consider the quadratic operator 1.5) or assume µ = 2). Let N = 2 and pose that the diffusion coefficients fulfill 1.2). Let a i satisfy 4 a i 1 + x + lna i ) ) dx = M <. 1.6) Then, 1.1) admits a global solution such that for any < T T <, a i belongs to L T, T ) ). Theorem 1.2 Let N = 1 with µ 3 and pose that the diffusion coefficients fulfill 1.2). Let a i satisfy 1.6). Then, for any < T T <, a i belongs to L T, T ) ). We point out that these statements do not require any regularity property on the diffusion coefficients D i which are only posed to be bounded. As a byproduct, we expect that such a result can be the starting point to apply the strategy developed in [9, 1] and then this would lead to the proof of the convergence to the equilibrium state for large time, with an exponential rate. Moreover, by using the new bound, a direct bootstrap argument combined to the standard local existence of smooth solutions shows the global regularity of the solution. Corollary 1.1 Let the assumptions of Theorem 1.1 or 1.2 be fulfilled. Suppose moreover that the D i s are of class C k ) with bounded derivatives up to order k. Then the solution belongs to L T, T ; H k )). Accordingly for C coefficients with bounded derivatives, the solution is C on T, T ). In higher dimensions, the same method provides information on the Hausdorff dimension definitions are recalled in Section 4) of the set of the singular points of the solutions. Theorem 1.3 Let N = 3 or N = 4 and µ = 2. We pose that the coefficients D i are constant with respect to x. Let a i satisfy 1.6). We consider a solution of 1.1) on, T). We call singular point any point t, x) having a neighborhood on which one of the function a i is not C. Then, the Hausdorff dimension of the set of singular points of the solution a does not exceed N 2 4)/N. 4

5 The existence of a weak solution in this case has been proved in [26]. In the next section, we briefly recall the fundamental estimate that follows from 1.4). This bound is used in Section 3 where we adapt De Giorgi s approach to the system 1.1). There, the restriction on the space dimension and the degree of nonlinearity will appear clearly, through the application of the Gagliardo-Nirenberg inequality. Section 4 is devoted to the estimate of the Hausdorff dimension of the set of singularities in higher space dimensions. 2 Entropy Dissipation In the following Sections, we adopt the viewpoint of discussing a priori estimates formally satisfied by the solutions of 1.1). As usual the derivation of such estimates relies on various manipulations as integrations by parts, inversion of integrals and so on. Then, they apply to the smooth solutions of the problem that can be shown to exist on a small enough time interval by using classical reasoning for non linear parabolic equations. Besides, these estimates also apply to solutions of suitable approximations of the problem 1.1). Such approximation should be defined so that the essential features of the system are preserved. Hence, let us reproduce the reasoning in [11]: by truncation and regularization we deal with an initial data a,η i C c ), a,η i which converges in L 1 ) to a i η> a,η i 1 + x + lna,η i ) ) dx C as η > tends to and such that a i 1 + x + lna i ) ) dx = C M <. Next, let us consider a cut-off function ζ Cc R) such that ζs) 1, pζ) B, 2) and ζs) = 1 for s 1. Then, in 1.1) we replace Q i a) by Q η i a) = Q ia)ζη a ), with a = a a 2 p. Accordingly, for any η > fixed, and a i L 1 ), Q η i a) belongs to L ). We can show that the corresponding regularized problem admits a unique smooth solution, globally defined, see [17, 28]. Therefore, in what follows we discuss a priori estimates on solutions of 1.1): for the sake of simplicity we detail the arguments working directly on 1.1), but we keep in mind that the arguments apply to the regularized problem as well. In turn, we obtain bounds on the sequence a η i, which are uniform with respect to η >. Finally, existence of a global solution satisfying the estimates follows by performing the passage to the limit η ; a detail that we skip here, referring for instance to [11]. As a warm up, let us discuss the a priori estimates that can be naturally deduced from 1.3) and 1.4). The results here apply in full generality, without assumptions on p, N, µ. 5

6 Proposition 2.1 Let a i satisfy ai 1 + x + lna i ) ) dx = M <. 2.7) We set p Dt, x) = a µ i i p ) a ν i i ln Then, for any < T <, there exists < CT) < such that { p a i 1 + x + lnai ) ) t t, x) dx + t T + p p t If is a bounded domain, this estimate holds for T = +. a µ i i a ν i i t, x). x ai 2 s, x) dx ds } Ds, x) dx ds CT). Proof. As a consequence of 1.3) and 1.4), we get d a i 1 + lnai ) ) dx + D i x a i xa i dx + dt a i Then, 1.2) allows to bound from below D i x a i xa i dx α a i x a i 2 dx = 4α In a bounded domain this is enough to conclude since we have a i lna i ) dx = a i lna i ) dx 2 a i a i lna i ) dx + p 2 e. D dx =. x ai 2 dx. a i lna i )1 ai 1 dx In the whole space, we compute, still by using 1.3) and denoting by M the norm of the diffusion coefficients, d x a i x dx = D i x a i dt x dx M α 2 x a i dx = M x a i 2 6 a i dx + M2 2α x a i ai ai dx a i dx.

7 Hence, we arrive at t a i 1 + x + lnai ) ) dx + α 2 4 t M + M2 a i dx ds 2α 1 + tm 2 /2α) ) M. x a i 2 a i dx ds + t D dx ds It remains to control the negative part of the a i lna i ) s; to this end, we use the classical trick a i lna i ) dx = a i lna i ) dx 2 a i lna i ) 1 ai e + 1 x /2 e x /2 a i 1) dx a i lna i ) dx + 4 e x /4 dx + x a i dx e since s lns) 2 e s for any s 1. We conclude readily by combining together all the pieces. 3 L bounds In the spirit of the Stampacchia cut-off method, L bounds of solutions of certain PDEs can be deduced from the behavior of suitable non linear functionals. Here, such a functional is constructed in a way that uses the dissipation property 1.4). Let us consider the non negative, C 1 and convex function { 1 + z) ln1 + z) z if z, Φz) = if z. Then, for k, we are interested in the evolution of Φa i k) dx. Lemma 3.1 There exists a universal constant C, such that for every a = a 1,...,a p ) solution of 1.1), for any k, and for any s t <, we have Φa i k)t, x) dx + 4α Φa i k)s, x) dx +C t t where [z] + = max, z) denotes the non negative part of z. s s x 1 + [ai k] + 2 τ, x) dx dτ 1 + k µ k)[a i k] µ 1 ) + ln1 + [ai k] + )τ, x) dx dτ 7

8 Remark 3.1 Notice that the universal constant does not depend on the actual solution a nor on k. It is also worth noticing that, in order to make sense of this inequality we need only a µ 1 i ln1 + a i ) to be integrable, although it is required to have a µ i to be integrable to make sense of the equation 1.1). This point will be very important in the next section. We got this feature thanks to the similarity of the function Φ and the natural entropy of the system 1.1). Proof. Multiplying 1.1) by Φ a i k) and summing yield d dt Φa i k) dx + D i x a i x a i Φ a i k) dx = Q i a)φ a i k) dx. 3.8) On the one hand, we observe that 1.2) leads to D i x a i x a i Φ a i k) dx = D i x a i x a i dx = D i x 1 + [a i k] + ) x 1 + [a i k] + ) 1 + [a i k] + x 1 + [a i k] + ) 2 α dx 1 + [a i k] + 4α x 1 + [ai k] + 2 dx. On the other hand, we rewrite the right hand side of 3.8) as = Q i a) ln1 + [a i k] + ) dx Qi a) Q i 1 + [a k] + ) ) ln1 + [a i k] + ) dx + Q i 1 + [a k] + ) ln1 + [a i k] + ) dx, 1 ai k 1 + [a i k] + dx where 1.4) implies that the last term is. We are thus left with the task of estimating Q i a) Q i 1 + [a k] + ) ) ln1 + [a i k] + ). To this end, let us consider the polynomial function P : R p R defined by Pu) = p uν i i. Clearly, given u, v R p, we have Pu) Pv) = 1 Pu + sv u)) u v) ds C u v 1 Pu + sv u)) ds where stands for any norm on R p. As a matter of fact, since the µ i s and ν i s are non-zero integers, we have j Pu) = ν j p uν i,j i where ν i,j = ν i if i j and ν j,j = ν j 1. In particular, 8

9 note that p ν i,j = µ 1. Therefore, working with the l 1 norm, we get Pu) p ) ν j u i ν i,j which yields, by using the convexity of the functions z z ν i,j, Pu) Pv) j=1 u l v l l=1 j=1 p p ) ν j u i ν i,j + v i ν i,j. Clearly, we have p u i ν i,j C p 1 + u i µ 1 ) and finally we obtain Pu) Pv) C u l v l ν j 1 + u i µ 1 + v i µ 1 ). l=1 We apply this inequality with u i = a i and v i = 1 + [a i k] + and we make use of the following simple remarks i,j=1 1 + [a i k] + ) µ 1 C 1 + [a i k] µ 1 + ), a i [a i k] + + k so that a µ 1 i C[a i k] µ k µ 1 ), a i 1 + [a i k] + ) 1 + a i [a i k] k. Applying the same reasoning with µ i replacing ν i, we arrive at Qi a) Q i 1 + [a k] + ) C 1 + k) j=1 1 + k µ 1 + [a j k] µ 1 ) +, where the constant C depends on µ and p <. Then, we end the proof by using the simple inequality: for any u, v, u µ 1 ln1 + v) + v µ 1 ln1 + u) 2 u µ 1 ln1 + u) + u µ 1 ln1 + u) ). As usual we have adopted the convention to keep the same notation C for a constant that does not depend on the solution, even when the value of the constant might change from one line to the other.) Remark 3.2 We point out that the arguments above do not apply so easily to situations where the unknown a is an infinite sequence, like e.g. for coagulation-fragmentation models the constant C involves a sum over the reactants). Let < T < T < and < K < be fixed. Set Let us denote U n = t n t T < t n = T1 1/2 n ) < T < T, < k n = K1 1/2 n ) < K. Φa i k n )t, x) dx + 4α 9 T t n x 1 + [ai k n ] + 2 τ, x) dx dτ.

10 The aim is to show that, for a suitable choice of K >, U n tends to as n which will yield the L bound. We start by making use of Lemma 3.1 with s t n t T and we average with respect to s, t n ). After some manipulations, we obtain U n 2n T +C tn T Φa i k n )s, x) dx ds ) 1 + k µ n k n )[a i k n ] µ 1 + ln1 + [ai k n ] + )τ, x) dx dτ. The crucial step consists now in establishing the following non-linear estimate, where restriction on both the space dimension N and µ appear. Proposition 3.1 Suppose N 2. There exists a constant C > which does not depend on the solution, nor on T, T, K) such that where U n C Kn, K, T) U N+2)/N n 1 Kn, K, T) = Q K T 2n2N+4)/N K µ )S K 2 nn+4)/n K)R K 2 n2n+4)/n µ) and S K = 2 max1/k N+4)/N, 1/K N+2)/N ), R K = 2 max1/k 2N+4)/N µ, 1/K 2N+1)/N µ ), Q K = S K + 2 max1/k 4/N, 1/K 2/N ). Let us explain how the restrictions on N and µ work. First of all, it will be crucial to remark that Kn, K, T) is bounded with respect to K > 1 provided µ 2N + 1)/N 1 = N + 2)/N which means µ = 2 in dimension N = 2 and µ = 2 or 3 in dimension N = 1. Second of all, we go back to Lemma 3.1 and we shall exploit the dissipation term that comes from the diffusion. Indeed, we expect an estimate of Φa i k) in L, T ; L 1 )) together with an estimate of 1 + [a i k] + ) 1/2 x 1 + [a i k] + ) in L 2, T ) ). Combining these information would lead to x Z[a i k] + ) L 2, T ; L 1 )) where Zu) = u Φz) 1 + z dz = u ln1 + z) z 1 dz. Let us consider a nonnegative function u defined on [T, T ] such that Zu) belongs to L T, T ; L 1 )) and x Zu) belongs to L 2 T, T ; L 1 )) According to the Gagliardo-Nirenberg-Sobolev inequality see [4], Th. IX.9, p. 162) the latter implies that Zu) L 2 T, T ; L N/N 1) )). we seek a homogeneous Lebesgue space with respect to the variables t, x. For N 2 we can obtain: Zu) L N+2)/N T, T ) ) )

11 Indeed, if N = 2 we have N + 2)/N = N/N 1), and if N = 1: T T v 3 dx dt vt) L 1 ) vt) 2 L ) dt v L T,T ;L 1 )) v 2 L 2 T,T ;L )). T T Eventually, we aim at comparing Zu) N+2)/N to ψu) ln1 + u) where ψ : R + R + has a polynomial behavior. Specifying the behavior of ψ will induce restrictions on µ that depend on the space dimension. Of course, it suffices to discuss the comparizon as u and u. Since ln1 + z) + 1/1 + z) 1 z z 2 /2 we first obtain that Zu) N+2)/N u 2N+4)/N /8 for u [, δ], δ > small enough. It follows that ψu) ln1 + u) can indeed be dominated by Zu) for bounded u s provided ψu) u u N+4)/N. Next, there exists A > such that for z A large enough, we have ln1 + z) + 1/1 + z) 1 1 ln1 + z). Thus, for u 2A we get 2 Zu) 1 2 u u/2 ln1 + z) dz u ln1 + u/2) C 1 u ln1 + u). Hence Zu) N+2)/N dominates ψu) ln1 + u) provided N 2 and ψu) u u N+2)/N. Reasoning the same way, we also prove that there exists C > such that Zu) C Φu) holds for any u. Let us summarize the properties that we need to justify Proposition 3.1. Lemma 3.2 Let us set There exists a constant C > such that ψu) = u N+4)/N 1 u 1 + u N+2)/N 1 u 1. ψu) ln1 + u) C Zu) N+2)/N, and Zu) C Φu). holds for any u. Furthermore, for every nonnegative function u defined on [T, T ] we have: T 2 ) T Zu) dx dτ Φu)τ, x) dx x 1 + u 2 τ, x) dx dτ. T T τ T Proof of Proposition 3.1. The proof splits into two steps: in the former we modify 3.9) so that in the latter we can make the dissipation terms appear by appealing to the Gagliardo-Nirenberg inequality. Step 1. The first step consists in showing the following inequality: U n C Kn, K, T) T T ψa i k n 1 ) ln1 + [a i k n 1 ] + ) dx dτ, 3.1) where the auxiliary function ψ has been introduced in Lemma 3.2. We start by noting that if a i k n k n 1, then a i k n 1 )/k n k n 1 ) 1. Therefore we can write for any α, β, 1 ai k n [ai k n 1 ] ) α + [ai k n 1 ] ) β + 1{kn a k n k i 1+k n 1 } + 1ai 1+k n 1 k n k n 1 n 1 2nα K α[a i k n 1 ] α + 1 ai k n nβ K β [a i k n 1 ] β + 1 ai k n

12 By using these simple estimates with α = N + 4)/N, β = N + 2)/N and α = N + 4)/N µ + 1, β = N + 2)/N µ + 1 respectively note that in both case α β), we are led to and 1 + k µ n) ln1 + [a i k n ] + ) 1 + K µ ) 2 nn+4)/n S K ψa i k n 1 ) ln1 + [a i k n 1 ] + ), 1+k n )[a i k n ] µ 1 + ln1+[a i k n ] + ) 1+K) 2 n2n+4)/n µ) R K ψa i k n 1 ) ln1+[a i k n 1 ] + ). Coming back to 3.9) yields U n 2n tn Φa i k n 1 ) dx ds T +C 1 + K µ )S K 2 nn+4)/n K)R K 2 n2n+4)/n µ)) T ψa i k n 1 ) ln1 + [a i k n 1 ] + ) dx dτ. The first integral in the right hand side can be dominated in a similar way using α = 4/N, β = 2/N); precisely, we have 2 n T tn 2n T Φa i k n ) dx ds tn 1 T 2n2N+4)/N Q K 1 + [a i k n ] + ) ln1 + [a i k n ] + ) dx ds T Therefore, we have proved from 3.9) that 3.1) holds. Step 2. Now, we go back to Lemma 3.2 so that 3.1) becomes U n C Kn, K, T) T ψa i k n 1 ) ln1 + [a i k n 1 ] + ) dx ds. Z[ai k n 1 ] + ) N+2)/N dx dτ. 3.11) Let us distinguish depending on the dimension N = 1 or N = 2 how we conclude by using the Gagliardo-Nirenberg-Sobolev inequality. For N = 2, using the Gagliardo-Nirenberg-Sobolev inequality and Lemma 3.2, we obtain U n C Kn, K, T) [ T + T 12 x Z[a i k n 1 ] + ) dx ) 2 dτ Φ[a i k n 1 ] + ) dx) 2 ds ]

13 Then, we use the second statement in Lemma 3.2 to obtain U n C Kn, K, T) [ τ T T + T Φa i k n 1 )τ) dx C1 + T ) Kn, K, T) U 2 n 1. For N = 1, we proceed as follows U n C Kn, K, T) C Kn, K, T) C Kn, K, T) T [ Φ[a i k n 1 ] + ) dx) 2 ds Za i k n 1 )t, ) 2 L ) t T T [2T + t T C Kn, K, T) 1 + T ) U 3 n 1. This ends the proof of Proposition 3.1. ) x 1 + [ai k n 1 ] + 2 dx dτ ] ) Za i k n 1 )t, x) dx dt Za i k n 1 ) dx Zai k n 1 + x Za i k n 1 ) ) 2 dx dt] t T ) 3 Φa i k n 1 ) dx ) 2 T Φa i k n 1 ) dx Finishing the proof of the L bound needs the following elementary claim. ] x 1 + [ai k n 1 ] + 2 dx dt Lemma 3.3 Let V n )n N be a sequence verifying V n M n V q n 1 for some M >, q > 1. Then for any n N, there exists ε, such that if V n lim n V n =. < ε, then Proof. Without loss of generality we pose n =. Let us set W n = lnv n ). We have W n n lnm) + qw n 1 which yields n W n lnm) q n j j + q n W q n lnm 1/q1 1/q)2) V ). j= 13

14 So, if V < M 1/q1 1/q)2), W n converges to, and V n converges to. Hence, it remains to check that the first term of the iteration can be made small choosing K large enough. Indeed, let us go back to Proposition 3.1. Picking K > 1, we can summarize the obtained estimate as U n C1 + 1/T) 2 n2n+4)/n U N+2)/N n 1. The keypoint is to remark that Q K, KR K and K µ S K remain bounded for large K s so that the constant C above does not depend on K. Hence, we apply Lemma 3.3 to V n = C1+1/T) ) N/2 Un. Now, let us specialize 3.11) to the case n = 2; we get with C which still does not depend on K) [ T 2 T ] 2 U 2 C x Za i K/2) + dx) dt + Φa i K/2) + dx ) dt 3.12) in dimension N = 2 and in dimension N = 1 the same expression is multiplied by the quantity p t T Φa i K/2) dx. This allows to establish the following statement. Lemma 3.4 Let ɛ >. Then, there exists K ɛ 1 such that for any K K ɛ we have U 2 ɛ. Proof. The proof reduces to prove that the two integrals in the right hand side of 3.12) tend to as K +. As a matter of fact, there exists C > such that for any z we have 1 + z) ln1 + z) C z1 + lnz) ). Furthermore, there exists C > such that for any k > 1 and z, we have [z k] ln[z k] + ) ) C z1 + lnz ). Accordingly, we deduce that Φa i K/2) converges to for a.e t, x), T) as K goes to infinity and is dominated by a i 1 + lna i ) ), which verifies T 2 a i 1 + lna i ) ) dx ) dt < owing to Proposition 2.1. Applying the Lebesgue theorem then shows that { T } 2 Φ[a i K/2] + ) dx) dt =. Next, we simply write lim K x Za i K/2) = 1 ai K/2 ln1 + [a i K/2] + ) [a i K/2] + 1 x a i. Then, we remark that z ln1 + z) + 1/1 + z) 1 is non decreasing which allows to evaluate x Za i K/2) 1 ai K/2 ln1 + a i ) x a i = 1 ai K/2 x Za i ) x Za i ). 1 + a i 14

15 We readily conclude since the 1 ai K/2 x Zu) decreases to as K for a.e t, x), T) while Lemma 3.2 yields T 2 x Za i ) dx) dt t T Φa i ) dx T x ai 2 dx dt < by using the basic estimates in Proposition 2.1 again. We conclude by classical integration theory arguments. We can now finish the proof of Theorem 1.1. Let us emphasize the dependence with respect to K by denoting U n K). We first fix K which makes U K) 2 small enough remark that K is more constrained as T is chosen small) so that we obtain by applying Lemma 3.3 lim n UK) n =. However, we clearly have U K) n 1 T t n T t n Φa i k n ) dx dt. Letting n go to infinity and applying the Fatou lemma, we deduce that 1 T T T T Φa i K) dx dt =, which implies that a i t, x) K for a.e t, x) T, T ). 4 Haussdorff Dimension of the Set of Singular Points In this section we study the Hausdorff dimension of the blow-up points of the solutions of 1.1). The derivation of the necessary estimates remains close to the strategy described in the previous section; again restrictions on the space dimension and the degree of nonlinearity appear. It turns out that relevant results can be obtained by this method in dimension N = 3 and N = 4 with µ = 2, while we are not able to reach improvements in direction of higher nonlinearities for lower dimensions. For the sake of simplicity, in what follows we assume that the diffusion coefficients D i are constant with respect to the space variable but they still depend on i, otherwise the problem becomes trivial by remarking that ρt, x) = p a it, x) satisfies the heat equation t ρ D x ρ =, with D the common value of the diffusion coefficients). Then, we shall prove Theorem 1.3. To begin with, let us recall a few definitions about Hausdorff dimension. For a given nonempty set A R d, s, δ >, we set { Hδ s A) = inf Γ1/2) s diamai ) )s, A 2 s Γs/2 + 1) i i } A i, diama i ) δ, 15

16 and then H s A) = lim δ Hδ s A). The Hausdorff dimension of A is defined by dim H A) = inf{s >, H s A) = } = {s >, H s A) = + }. We refer to [13] p. 171) for more details. The starting point of the proof is two-fold: first we identify the highest L p norm available with the estimate deduced from mass conservation and entropy dissipation, and second we remark that the problem admits an invariant scaling. This is the purpose of the following claims. Lemma 4.1 Let N > 2. There exists C > such that for any u L, T; L 1 )) verifying u L 2, T) ), we have T ) N u N+2)/N N+2 dx dt ) C 1 + T) u L,T;L 1 )) + x u 2 L 2,T) ). Lemma 4.2 Let a be a solution of 1.1). Let t > and x. Then, for any < ε 1 satisfies 1.1). a ε t, x) = ε 2/µ 1) at + ε 2 t, x + εx) Keeping in mind Lemma 4.2 we consider now solutions of 1.1) that are defined for negative times. Let us set k n = 1 1/2 n, t n = 1 + 1/2 n B n = B, t n ), Q n = t n, ) B n. Note that B n B n 1 and Q n Q n 1. We introduce a cut-off function ζ n : R N R, ζ n x) 1, ζ n x) = 1 for x B n, ζ n x) = for x B n 1, ijζ 2 n x) C 2 2n. i,j {1,...,N}, x R N We define U n = t n t B n Φa i k n ) dx + Q n x 1 + [ai k n ] + 2 dx ds. Multiplying 1.1) by ζ n x)φ a i k n ) we obtain the following localized version of 3.8) d dt = Φa i k n ) ζ n dx + Q i a)φ a i k n )ζ n dx + D i x a i x a i Φ a i k n ) ζ n dx D i : D 2 ζ n Φa i k n ) dx, 4.13) 16

17 where D 2 ζ n stands for the hessian matrix of ζ n and A : B = N k,l=1 A klb kl. Remark that 1 Bn x) ζ n x) ζ n 1 x) 1 and 2 kl ζ nx) 2 2n 1 Bn 1 x). Then, reproducing the proof of Lemma 3.1 and 3.9) we obtain ) U n C 2 2n + 2n p Φa i k n )s, x) dx ds T Q n 1 ) +C 1 + k µ n k n )[a i k n ] µ 1 + ln1 + [ai k n ] + )τ, x) dx dτ. Q n 1 From this relation we shall be able to establish the following statements. Proposition 4.1 Let N > 2. The following relation holds U n C 2 nµ+1) U 1+2/N n 1 for any n 1. Accordingly, if U is small enough then lim n U n =. 4.14) Corollary 4.1 There exists a universal constant η > such that any solution of 1.1) satisfying is such that for any i {1,..., p} we have 3 B,3) a i N+2)/N dx dτ η a i t, x) 1 a. e. in 1, ) B, 1). Proof of Proposition 4.1. We start the proof by remarking that k n 1, and k n k n 1. Consequently we have [a i k n ] + [a i k n 1 ] +. Then, we make a i k n 1 appear by using ) µ 1 ai k n 1 1 ai k n 1 k n k ai kn 1 2 nµ 1) [a i k n 1] µ 1 +. n 1 It follows that the last integral in 4.14) can be dominated by C1 + 2 nµ 1) ) Similarly, we check that Q n 1 [a i k n 1 ] µ 1 + ln 1 + [a i k n 1 ] + ) dx dτ. Φa i k n ) 1+[a i k n ] + ) ln 1+[a i k n ] + ) 2 nµ 1) +2 nµ 2) )[a i k n 1 ] µ 1 + ln 1+[a i k n 1 ] + ). Combining these inequalities, and identifying the highest power of 2 n involved we finally get from 4.14) U n C T 2 nµ+1) [a i k n 1 ] µ 1 + ln ) 1 + [a i k n 1 ] + dx dτ. 4.15) Q n 1 17

18 The next step consists in evaluating u µ 1 ln1 + u) by functions having a polynomial behavior, so that the right hand side of 4.15) will be dominated by a power larger than 1 of U n 1, up to the use of the Gagliardo-Nirenberg-Sobolev estimate. To be more specific, we note that there exists C > such that for any u, on the one hand and, on the other hand u 2 1 u 1 + u 1 u 1 C Φu) u µ 1 ln1 + u) Cu µ 1+α 1 u 1 + u µ 1+α 1 u 1, α 1, < α. The choice of the exponents α, α comes from information in Lemma 4.1 which then leads to the constraints N {3, 4}, and µ = 2. Hence, with these considerations 4.15) becomes U n C T 2 nµ+1) [a i k n 1 ] µ 1+α + 1 ai k n 1 1 Q n 1 ) +[a i k n 1 ] µ 1+α + 1 ai k n 1 1 dx dτ { ) 2/N } C T 2 nµ+1) Φa i k n 1 ) dx x 1 + [ai k n 1 ] + 2 dx dτ t n t B n 1 Q n 1 C T 2 nµ+1) U 1+2/N n 1. Coming back to Lemma 3.3 finishes the proof of Proposition 4.1. Proof of Corollary 4.1. We are thus now left with the task of discussing the smallness of U = 2 t B,2) Φa i ) dx + 2 B,2) x ai 2 dx dτ. It turns out that this quantity can be evaluated by involving the norm L N+2)/N on a slightly larger domain; this is precisely the purpose of Corollary 4.1. The mass conservation yields d dt ζ x)a i t, x) dx = D i : D 2 x ζ x) a i t, x) dx C B,3) a i t, x) dx, since ζ has bounded second order derivatives and is ported on B, 3). We integrate over the time interval t, τ), and then we average over τ 3, 2). Hence, we get 2 t B,2) a i t, x) dx C C B,3) B,3) B,3) a i τ, x) dx dτ a i τ, x) dx dτ a i τ, x) N+2)/N dx dτ) N/N+2).

19 Similarly, the entropy dissipation yields d dt = C ζ x) a i lna i ) dx + D i : D 2 x ζ x) a i lna i ) dx B,3) a i lna i ) dx. ζ x) ζ x) D i x a i x a i ) a i Again we integrate with respect to the time variable. We shall also use the trick u lnu) = u lnu) 2u lnu)1 u 1 u lnu) + C u. dx It follows that 2 t C B,2) a i lna i ) dx + α 3 B,3) 2 B,2) a i lna i ) + a i ) dx dτ. x ai 2 dx dτ Furthermore, we have u lnu) = u lnu)1 u 1 u lnu)1 u 1 C u N+2)/N + u). Therefore, using the Holder inequality, we end up with 2 t B,2) a i lna i ) dx C Coming back to the definition of U finishes the proof. ) a i N+2)/N L N+2)/N 3,) B,3)) i 1/2 L N+2)/N 3,) B,3)). Now, these statements allows us to deduce some property of the solution of the original Cauchy problem by using the scaling argument in Lemma 4.2. Indeed, we notice that 3 B,3) t +3ε2 a ε τ, x) N+2)/N dx dτ = ε 2N+2)/N N+2) t 3ε 2 y x 3ε as, y) N+2)/N dy ds holds recall that we are dealing with the case µ = 2 only). We deduce the following statement. Lemma 4.3 Let N = 3 or N = 4 and µ = 2. Then there exists a universal constant η > such that for any a solution of 1.1), any t >, x and < ε 1, we have the following property. If: 1 t +3ε 2 as, y) N+2)/N dy ds η ε N+2 ε 2N+2)/N t 3ε 2 y x 3ε then a i satisfies a i t, x) 1/ε 2 on t t ε 2, x x ε and a i is C on this set. 19

20 Notice that it is enough to show the boundedness of the a i. Then the full regularity on the possibly smaller) neighborhood is obtained by induction, using classical theory of parabolic equations. We start by localizing: namely, we consider, T) B, R), < T, R <. We set S = {t, x), T) B, R), u is not C on a neighborhood of t, x)}. We cover S by rectangles with step size ε 2 in the time direction and ε in the space directions, centered at points t, x) S. By the Vitali covering lemma see [32], p. 9) we actually consider a countable covering of disjoints such rectangles at the price of multiplying ε by a suitable constant which does not matter). We denote {C j, j N} the set of these rectangles, centered at t j, x j ) S. Since t j, x j ) denies the conclusion of Lemma 4.3, we have 1 ε N+2 C j a i s, y) N+2)/N dy ds η ε 2N+2)/N where C j stands for the rectangle centered at t j, x j ) with step size 3ε 2 in the time direction and 3ε in the space directions. We introduce the function 1 F S t, x) = 1 C j t, x) a ε N+2 i s, y) N+2)/N dy ds. C j Hence, denoting by L the Lebesgue measure, we have the following estimate ) L j N C j L t, x), T) B, R), F S t, x) η /ε N+2)/N) T F S t, x) dx dt εn+2)/n η as a consequence of the Tchebyschev inequality. It yields by direct evaluation T ) L j N C j ε2n+2)/n 1 η a i N+2)/N C j t, x) dx dt dy ds C j L C j ) = ε2n+2)/n η j N a i N+2)/N L N+2)/N,T) ). Since the Lebesgue measure of the C j s is proportional to ε N+2, we deduce that the cardinal of the covering is of order Oε 2N+2)/N N+2) = ε N2 4)/N ). Furthermore, the C j s realize a covering of S with sets of diameter ε; we conclude that the Hausdorff dimension of S is dominated by N 2 4)/N. Remark 4.1 It is not obvious that we can improve this estimate, which is in the spirit of [29, 3] for the Navier-Stokes equations, up to a sharp result as in [5, 18]. A difficulty is related to the fact that we are dealing with diffusion coefficients that depend on the component of the system, which prevents from using regularizations by a common heat kernel. 2

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