Conservative cross iffusions an pattern formation through relaxation Mostafa Benahmane, Thomas Lepoutre, Americo Marrocco, Benoît Perthame To cite this version: Mostafa Benahmane, Thomas Lepoutre, Americo Marrocco, Benoît Perthame. Conservative cross iffusions an pattern formation through relaxation. Analyse mathématique et numérique e sytèmes cross-iffusion. 29. <inria-368595> HAL I: inria-368595 https://hal.inria.fr/inria-368595 Submitte on 17 Mar 29 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.
Conservative cross iffusions an pattern formation through relaxation Mostafa Benahmane Thomas Lepoutre Americo Marrocco Benoît Perthame 13th March 29 Abstract This paper is aime at stuying the formation of patches in a cross-iffusion system without reaction terms when the iffusion matrix can be negative but with positive self-iffusion. We prove existence results for small ata an global a priori bouns in space-time Lebesgue spaces for a large class of iffusion matrices. This result inicates that blow-up shoul occur on the graient. One can tackle this issue using a relaxation system with global solutions an prove uniform a priori estimates. Our proofs are base on a uality argument à la M. Pierre which we exten to treat egeneracy an growth of the iffusion matrix. We also analyze the linearize instability of the relaxation system an a Turing type mechanism can occur. This gives the range of parameters an ata for which instability may occur. Numerical simulations show that patterns arise inee in this range an the solutions ten to exhibit patches with stiff graients on boune solutions, in accorance with the theory. 1 Introuction The ynamics of interacting population with cross-iffusion have been wiely investigate by several researchers. The concept of this phenomena was stuie by Levin [18], Levin an Segel, [17], Okubo [27], Mimura an Murray [24], Mimura an Kawasaki [23], Mimura an Yamaguti [25], an many other authors. All these papers base the pattern formation on a reaction term as prey-preator interactions. Spatial patterns can however emerge from pure iffusions without reaction terms nor oriente rift at the iniviual level. This is the case of N populations escribe microscopically by a brownian process which intensity epens upon the macroscopic ensity U = (U 1,..., U N ) of the populations X k (t) = σ k ( U(X, t) ) Wk (t), 1 k N. Departamento e Ingeniería Matemática Faculta e Ciencias Físicas y Matemáticas, Universia e Concepción, Casilla 16-C, Concepción, Chile Email: mostafa benahmane@yahoo.fr INRIA-Paris-Rocquencourt, Team BANG, Domaine e Voluceau, BP15, 78153 LeChesnay Ceex, France. Emails: thomas.lepoutre@inria.fr Université Pierre et Marie Curie-Paris 6, UMR 7598 LJLL, BC187, 4, place Jussieu, F-75252 Paris ceex 5 Email: americo.marrocco@inria.fr Institut Universitaire e France. Email: benoit.perthame@upmc.fr 1
When set on a boune omain with reflexion on the bounary, the corresponing moels for the population ensity are cross-iffusions { tu A(U) =, in, (1) n A(U) = on, where U = U(x, t) R N, x a smooth boune omain of R, n enotes the outwar normal to. Finally A : R N R N is a nonlinearity relate to the intensity of the interactions by the relation A k (U) = U k a k (U), (2) an a k (U) = 1 2 σ k(u).σk t (U). We also complete the system with an initial ata U(t = ) = U = (U 1,..., U N ) with U k. The properties, an pattern formation capacity of such systems are better escribe by introucing the more general form N t U k iv[d kl (U) U l ] =, (3) l=1 where D kl (U) are the components of a N N matrix, the erivative of A in the case (1). Bounary conitions have to be impose an we consier here the case of Neumann conitions D(U). U.n =, on. For such bounary conitions, mass conservation yiels naturally U(t) = U, t, where U enotes the average U = 1 U(x)x. The Lotka-Volterra competition with cross-iffusion has recently receive great attention. They are many establishe results concerning the global existence of classical solutions (see [33, 19] an the references therein) where most of the proofs rely on Amann s theorem [1, 2]. We point out that stanar parabolic theory is not irectly applicable to our moel ue to the presence of cross-iffusion terms. In opposition with pattern formations, an important issue has been wiely stuie which is to know in which circumstances the solutions exist globally an behave like in the case a single heat equation, i.e., relax to a constant state as t. Typically three kins of special methos have been helpful in this scope. The first metho is to rely on the maximum principle. It can occur on certain combinations of the U i as in [1, 2, 3]. Entropy methos also applies to particular systems an has also been a useful tool because of the relate symmetrization of the system following [15, 8]. It provies a natural metho both for existence an relaxation to steay state as in the recent stuies in [6, 7] of the Shigezaa-Kawasaki prey-preator system [31], or for tumous moels [12]. This metho typically applies in the special case of the square entropy when D is efinite positive meaning that there is ν > such that N ξ k D kl (U)ξ l ν ξ 2, ξ R N. (4) k,l=1 2
This strong positivity property gives the energy inequality 1 2 t U(t) 2 x = N k,l=1 U k D kl (U) U l ν U(t) 2 x. The thir metho, by uality, has been use in [5] on a particular upper-iagonal iffusion with Dirichlet bounary conition; we show here that the metho can be extene to very general systems with Neuman conitions. Concerning instabilities, the interplay between iffusion an reaction terms has raise surprising results in the spirit of the Turing instability mechanism [32]. The question to know if cross-iffusion or self-iffusion gives an avantage to competing species is stuie in [2, 21]. Our interest in this paper concerns instability mechanisms that may appear only from the iffusion intensity an the lost of positivity in the secon orer matrix. We stuy in which circumstances the increase of this intensity with higher ensity of the other species can lea to a segregation phenomena. Of course such instability is incompatible with any entropy inequality, an thus (4) cannot hol. This rises several mathematical questions which seem to be new in the omain of cross-iffusions. Can it still be that small solutions exist globally even though the maximum principle oes not hol in general? For large ata, what kin of regularity or blow-up can we expect? Finally, how o regularize systems behave in the instability regime. We will stuy these questions with a moel problem in min relate to Shigezaa-Kawasaki s system an that represents two species with stronger interactions { t U 1 [U 1 (1 + a 11 U p 1 + a 12U p 2 )] =, x, (5) t U 2 [U 2 (1 + a 21 U p 1 + a 22U p 2 )] =, still with Neumann bounary conitions. One can check that as soon as p > 1, the matrix D kl is negative for U large. We approach these questions both theoretically an numerically. In particular we prove existence for small initial ata (section 2) an we give a priori bouns in L p t,x for possible solutions to (5) thus showing that the break-own shoul come from the blow-up of graient estimates rather than usual L p norms (section 3). Our main tool here is a general estimate ue to M. Pierre [29, 28] in the context of semilinear parabolic systems (arising in population ynamics or more generally reaction-iffusion systems) (see also [9]). In section 4, these bouns are extene to a relaxation system that takes into account a local measurement of ensities; we show that the metho is well aapte to general (even not parabolic) cross-iffusions an prove global existence for the relaxation system. For non-parabolic cases, we show that Turing instability occurs in a certain range of ata an for small relaxation parameters. Numerical simulations of this relaxation system are performe in section 5. They show that the oscillatory initial regime reorganizes to create patches where one species ensity ominates the other an interfaces are generate which with is relate to the relaxation length. The technical an general extension of M. Pierre s estimate to boune omains for Neumann bounary conitions is kept for an Appenix as well as another remarkable energy estimate which hols for particular cross-iffusion coefficients in (5). 2 Global solutions for small ata The lack of maximum principle for general iffusion systems is a major ifficulty that arises for systems as (3). Using stronger H 1 estimates, we can show in one imension that for small initial ata there 3
is global existence. Such solutions ecay to the constant state for large time an this is incompatible with the patterns formation we are intereste in. This inicates that large initial ata are necessary for pattern formation as expecte in general. We consier the system uner the form (1) (2) an assume that a k C 1 (R N ), (6) a k () ν >. (7) Theorem 2.1 (Global small solutions in 1 imension) In one imension, uner assumptions (6),(7), an for an initial ata satisfying, with α small enough, U L () + U L 2 () α, there is a global solution U(t, x) to the cross-iffusion system (1) (2). It satisfies for all t > with C inepenent of t an U(t) L () Cα, U(t) L 2 () α, U(t) t U. Proof. Firstly, since U(t) is a priori conserve, we notice that U(t) L () U L () + U L 2 () α + U L 2 (). Seconly, we multiply (1) by the vector A (U) an ifferentiate. We obtain t A(U) = ( A (U) t U ) = ( A (U) A(U) ), we multiply by A(U) an integrate A(U) 2 = A(U) ( A (U) A(U) ) = t 2 A(U) A (U) A(U). (8) But from (2) an (7), we have A () kl = a kk ()δ kl, an thus for U L () ε small enough, we have XA (U)X ν 2 X 2, X R N, (9) We now consier T efine by A (U)X 2 ν2 2 X 2, X R N. (1) T := sup{t, U(t) ε}. For α < ε, then T >. Suppose T <, then for t T, we have, from (8) an (9), A(U(t)) 2 x A(U ) 2 x. Therefore, A ( U(t) ) U(t) 2 4 A(U ) 2 x.
Using (1), this leas to U(t) 2 x 2 ν 2 A(U ) 2 x 2C ν 2 U 2 x. Now, we choose α such that α ε 2C 3 an α ε ν 3, this ensures, U(t) L () 2ε 3 an thus, T is not maximal. Therefore T =. Finally, the existence of a solution for small times follows from stanar parabolic theory an the a priori boun above shows that these are global solutions. We now prove the time convergence to U. Because (U U ) 2 = U 2 U 2, we compute U ) t (U 2 = U 2 = A(U)U = A (U) U. U. t But as the solution stays in the omain U(t) L () ε, we have for some constant C, U t (U ) 2 ν U 2 2C (U U ) 2, 2 thanks to the Poincaré Wirtinger inequality. We conclue from Gronwall lemma that U U L 2 () e C t U U L 2 (). 3 A priori bouns for large ata For large initial ata an when the conition (4) oes not hol, we cannot expect in general the existence of solutions for the cross-iffusion system (3) (2). For a single equation, the corresponing situation is when A (u) can be negative on some interval I R +. u A(u) =. t The situation is analyze in [3, 26, 11] (see also the survey in [1]) an it is better analyze in term of relaxation systems, an approach which we will follow later. We expect that oscillations or jumps occur at positive times, but a first issue is a priori control in L for possible solutions. This follows from the maximum principle for a single equation (an possibly from entropy constructions for relaxation systems, see [3]). For systems this is an open question an we give here a first a priori boun Theorem 3.1 Smooth solutions to (3) (2) satisfy the a priori bouns ( T N N ) 1 2 A k (U) U k xt C1 () U ( N L 2 () + C 2, Uk ) T. (11) k=1 Therefore if we assume in (2) that k=1 then we also have, with the notation Q T = [, T ], a k (U) ν > k = 1,..., N, (12) ν U L 2 (Q T ) C 1 () U L 2 () + C 2 (, U ) T. (13) k=1 5
In the particular case of moel (5), we observe that the larger is p, the best is the boun in (11). In particular, A(U) is always integrable. The L 2 estimate in (13) is much weaker. Proof. Our proof is base on a variant of a general uality argument ue to [28, 9], that is presente in Appenix A. We enote w = N U k. We sum up the equations an we fin which we write k=1 t w N a k (U)U k =, k=1 t w α(t, x)w =, α(t, x) := We can use now (27) in Appenix A an obtain, N k=1 a k(u(t, x))u k (t, x) w(t, x) = α(u(t, x)). α w L 2 (Q T ) C() w L 2 () + 2 w α L 2 (Q T ). (14) In orer to control the right han sie by α w L 2 (Q T ), we use a truncation metho. Since the coefficients a k (U) are continuous, an U k are nonnegative, we may efine for any R >, sup α(u) := M(R) < +. w R Furthermore, we may truncate w away from values less than R, a parameter to be fixe later on, with the inicator function 1I {w R} an rewrite (14) as α w 1I {w R} L 2 (Q T ) C() w L 2 () + 2 w α1i {w R} L 2 (Q T ) + 2 w α 1I {w R} L 2 (Q T ), α w 1I {w R} L 2 (Q T ) C() w L 2 () + 2 w α w R 1I {w R} L 2 (Q T ) + 2 w M(R) L 2 (Q T ). We choose R = 4 w an obtain α w 1I {w R} L 2 (Q T ) 2C() w L 2 () + 4 w T M(R). Since we also know that αw1i {w R} L 2 (Q T ) R M(R) T = 4 w M(R) T, we conclue α w L 2 (Q T ) 2C() w L 2 () + C 2 ( w ) T, with C 2 ( w ) = 8 w M(4 w ). This is exactly the a priori estimate (11). The other statement is a simple an irect consequence. 6
4 A relaxation system If we assume that the intensity of the brownian motion epens on the ensity of the populations measure with a space scale δ > an not at the exact location x, then the system (5) can be replace by a cross-iffusion relaxation system { t u k [a k (ũ)u k ] =, x, k = 1,..., N, δ 2 ũ k + ũ k = u k, (15) together with Neumann bounary conitions both on u k an ũ k. Relaxation proceures are usual an several other examples for cross-iffusions can be foun in [14, 4] an for phase transitions see [13, 1, 3]. In terms of the ecological interpretation, it is also more realistic than the initial system (5), because iniviuals are unlikely to be able to access a pointwise ensity, but might estimate their environment from sensing at a smaller scale. We can expect that the system (15) is well-pose, an we first stuy this question. Then we prove uniform bouns inepenent of δ which inicates that instability shoul arise from the blow-up of graients. To tackle the question of instabilities, we show that the system exhibits Turing patterns for δ small, this is our secon goal in this section. We keep in min the example (5) an assume that for some p > one has < ν a k (U) C (1 + U p ), k {1,..., N}. (16) For later purpose, we also introuce the assumption that for some constant K > an some η >, we have a k (U) a η k (U) K U. (17) We have in min coefficients of the form (1 + Ũ p p 1 j ) an then we can take η = p. 4.1 Uniform estimates for p < 2 We first exten the a priori estimate of section 3 to this relaxation system. The coupling inuces a limitation on the possible growth of the nonlinearities a k (U) an we have the Theorem 4.1 Assume that (16) hols for some < p < 2, then, the a priori boun hols for a constant C inepenent of δ u L 2 (Q T ) C( u L 2 (), T ), T >. This is weaker than the a priori boun in (11). The ifficulty in the case at han comes from the epenency of a k (ũ) which we cannot lower boun from u itself. Proof. We enote by ã k the quantity a k (ũ). The estimate (27) of Appenix A gives, for all k {1,..., N} ã k u k L 2 (Q T ) C() u k L 2 () + 2 u k ã k L 2 (Q T ). 7
The last term may be estimate as T T ã k 2 L 2 (Q T ) = ã k C (1 + l ũ p l ). Thanks to Holer inequality an irect estimate on the solution to the elliptic equation on ũ l, we have, T ũ p l ( T ) 2 p 2 ũl p 2 p 2 L 2 (Q T C()T 2 ul p ) L 2 (Q T ). Finally, back to the original inequality we arrive at ν uk L 2 (Q T ) C() u k L 2 () + 2 C T u k + 2 u k C ( T ) 2 p p 4 u 2 L 2 (Q T ), which leas to ν u L 2 (Q T ) C(, u L 2 (), T ) + C(, u )T 2 p 2 u p 2 L 2 (Q T ). As p/2 < 1, this proves that u L 2 (Q T ) is a priori boune as by a constant epening only on, T, u L 2 () an the two constants in (16). 4.2 Existence of solutions We now show stronger estimates from which strong compactness of solutions follows. They use funamentally the regularity on ũ in (15) by elliptic regularizing effects. Existence of global solutions follow an the etails are carrie out in [16]. The main result of this section is the following Proposition 4.2 Assume that (16) hols with p > 1, an p < when > 2. Then, the a priori estimate hols ã i u i L 2 (Q T ) C(δ, u L 1 L 2 (), T ). (18) Furthermore, if we assume (17) in imension 1 with any η >, an in imension 2 with < η < 1, then we have for all 1 q <, 2 2 u(t) L q () C(q, δ, u L 1 L q (), T ), t T, (19) T u q/2 2 xt C(q, δ, u L 1 L q (), T ). (2) Proof. We begin with the proof of (18) which improves that of the theorem 4.1. We use again the estimate (27) applie to u i which yiels ν ui L 2 (Q T ) ã i u i L 2 (Q T ) C() u i L 2 () + 2 u i ã i L 2 (Q T ). (21) We use the hypothesis (16) to get T ã i L 2 (Q T ) C T + C ũ p C () T T + 8 ũ p = C ( T T + ũ p p).
Thanks to elliptic regularity we also have ũ p C(δ, r) u r, for any r > 1 satisfying also 1 p 1 r 2 (particularly it is true for any r if = 1, 2). Then, using interpolation inequality an choosing r < 2, we fin successively ũ p C(δ, r) u 1 θ(r) 1 u θ(r) L 2 () = C(δ, r) u 1 θ(r) 1 u θ(r) L 2 () θ(r) = 1 1 r 1 1 2 = 2(1 1 r ), ã i L 2 (Q T ) C ( T + T ã i L 2 (Q T ) C ( T + C(δ, r) p/2 u (1 θ(r))p/2 1 C(δ, r) p u (1 θ(r))p 1 u pθ(r) L 2 ()), T Now, if we may choose r such that θ(r)p < 2, we get, thanks to Jensen s inequality T which we rewrite as u(t) pθ(r) L 2 () t = T Replacing in (22), we have T An replacing in (21), we obtain u(t) pθ(r) L 2 () t). (22) ( T ) pθ(r)/2 u(t) 2(pθ(r)/2) t T 1 pθ(r)/2 u(t) 2 L 2 () L 2 () t, u(t) pθ(r) L 2 () t T 1 pθ(r)/2 u pθ(r) L 2 (Q T ). ã i L 2 (Q T ) C T + C(δ, r, p, u 1 )T 2 pθ(r) pθ(r) 4 2 u L 2 (Q T ) ν u L 2 (Q T ) C() u L 2 () + C () T + C(δ, r, p, u 1 )T 2 pθ(r) pθ(r) 4 u 2 L 2 (Q T ) (23) This conclues the first inequality when pθ(r)/2 < 1, an it remains to fin the range of p in orer to fulfill the constraints. These can be obtaine choosing r close enough to 1 for = 1, 2. For > 2, we nee the conitions { 1 p 1 r 2, 1 < r < 2, pθ(r) 2 = p(1 1 r ) < 1. We choose to satisfy the secon line 1 r > p 1 p it a posteriori). This leas, in the first line, to the conition p <, but close to equality (which gives 1 < r < 2 as we check 2 2, but close to equality (which imposes r < 2 +2 ). The boun on u L 2 (Q T ) gives then a boun on ã i u i L 2 (Q T ) thus concluing the proof of (18). This estimate leas to the stronger a priori bouns (19) that we prove now. We go back to the equation an write u q + C q ã u q/2 2 = Cq 1 u q/2 u q/2 ã. t 9
From this equality, an writing ã u q/2 = (ã 1/2 u q/2 ) u q/2 ã 1/2, we erive irectly the inequality t u q + Cq 3 ã u q/2 2 + Cq 3 (ã 1/2 u q/2 ) 2 Cq 2 u q ã 2 C q u q ũ 2 ã 2η 1. ã (24) We show separately how in imensions 1 an 2 this allows us to control any L q norm for q < +. The case = 1. The proof is easier for = 1 because there exists a constant C (epening only on δ an ) such that ũ L () + ũ L () C u C( u 1 ). Therefore from (24) we euce t u q C.C q We conclue thanks to Gronwall lemma. The case = 2. In imension 2, we ivie the proof into two steps. Step 1, 1 < q < 2. We first focus on small values of the exponent q namely 1 < q < 2 an 2η < 1 + q 2 (the limitation η < 1 comes from choosing q close to 2). From (24) an using successively (17) an Höler inequality, we obtain t u q C 3 q u q. u q ã q/2 ũ 2. Cq 3 u ã q qr ũ 2 2r thanks to elliptic regularity, with Choosing r = 2 q then 1 m = 1 q 4 t C(q, δ, r) u ã q qr u 2 m 1 m = 1 2r + 1 2 = 1 1 2r. an we arrive at u q C(δ, r) u ã q 2 u 2 m C(δ, r) u ã q 2 u 2θ q u 2(1 θ) L 2 () C(δ, r) u ã q 2 u q q u 2 q? L 2 () because, as we have 1 2 < 1 m < 1 q, we may interpolate m between q an 2 with θ = 1 m 1 2 1 q 1 2 = 2 q 4 2 q 2q = q 2. We finally obtain by Young s inequality, [ u q C(q, δ, r) u ] ã 2 2 t + u 2 L 2 () u q q, 1
an we may then conclue with Gronwall lemma using the estimates on T u ã 2 2 an T u 2 2 in (18). By interpolation, this also gives a priori boun for any L q norm for q [1, 2[. This ens step 1. Step 2 We now focus on L q norms for q 2. We notice that now, controlling any L q norm for q < 2, we control by elliptic regularity any L q norm of ũ except the L norm. We also control the L norm of ũ an therefore of ã. We go back to (24) an conclue t u q + C u q/2 2 C u q ã 2 ã C(δ, q, T ) u q ũ 2 C(δ, q, T ) u q qr ũ 2 2r C(δ, q, r, T ) u q qr. We use now interpolation: for any s > r > 1, we have u qr u 1 θ q u θ qs θ = 1 qr 1 qs 1 q 1 qs = 1 r 1 s 1 1. s Using Young s inequality, we obtain for ε small (to be chosen later) u q + C u q/2 2 C(δ, q, r, T, ε) u q q t + ε u q qs. We have by Poincaré Wirtinger inequality u q qs = u q/2 2 2s 2 u q/2 u q/2 2 2s + 2 uq/2 2 2s C(, s, ν) uq/2 2 + C(, s) u q q/2 C(, s, ν) uq/2 2 + C(, s) [ u q 1 + u q q], from interpolation an Young s inequality. We fix s as above, choose ε small enough an we obtain u q C(, q) [ u q 1 t + q] u q. An we conclue (19) by Gronwall lemma. The last estimate (2) also follows from (24). 4.3 Turing patterns In orer to go further an stuy the instability occurring in the regularize moel, we consier the following particular system: t u ( u(1 + ṽ 2 ) ) =, t v ( v(1 + ũ 2 ) ) =, δ 2 ũ + ũ = u, δ 2 ṽ + ṽ = v, still with Neumann bounary conitions an initial ata u, v. (25) 11
It is rather intuitive that for δ large, iffusion is ominant; this is also the case for small initial ata thanks to the argument in section 2. Therefore, the appearance of patterns epens upon a relation between the average ensities of populations u an v an the parameter δ. In orer to stuy this in etails, we begin with some notations the only possible constant steay state of the system is given by u = ũ = u an v = ṽ = v, we enote by (λ >, w) the non-zero solutions to the Neumann eigenproblem w = λw, n w = on we also enote by λ 1 the first eigenvalue for the Laplacian. In orer to investigate when the (in)stability of the constant steay state occurs, we stuy the linearize system: t u (1 + v 2 ) u 2 u v ṽ =, t v (1 + u 2 ) v 2 u v ũ =, δ 2 ũ + ũ = u, δ 2 ṽ + ṽ = v. As usual, we look for solutions of type e µt (a, b, c, )w. Such solutions shoul satisfy c = a 1+δ 2 λ, = b 1+δ 2 λ, µa + λa(1 + v 2 ) + λ2 u v b 1+δ 2 λ =, µb + λb(1 + u 2 ) + λ2 u v a 1+δ 2 λ =, which may be written uner the matrix form 1 + v 2 2 u v 1+δ 2 λ 2 u v 1+δ 2 λ 1 + u 2 We enote by M = M( u, v, δ, λ) this symmetric matrix. ( ) ( ) a = µ a. b λ b The question of the stability of the constant steay state can now be formulate in terms of eigenvalues of the matrix M. It is unstable if µ > an thus if M has negative eigenvalues. In this case, local behavior aroun the equilibrium shoul lea to segregation since the associate eigenvector to µ/λ shoul satisfy a.b <. We have the following Lemma 4.3 If the initial populations u an v are large enough an the relaxation parameter δ is small enough, then the constant steay state is linearly unstable. More precisely, it occurs uner the conitions γ := 4 u 2 v 2 ( 1 + u 2)( 1 + v 2) >, (1 2 u v + u 2)( 1 + v 2) δ 2 < (1 λ 1 + u 2)( 1 + v 2). (26) 12
The omain influences instability only through the smallness conition on δ 2 λ 1 () when the initial ata are such that γ >. Notice that the first conition ensures that the limiting system (δ = ) has a negative iffusion matrix D in the setting (3). Proof. As mentione earlier, the constant steay state is unstable if the symmetric matrix M amits negative eigenvalues, i.e., if et(m) <. We calculate et(m) = 4 u 2 v 2 (1 + δ 2 λ) 2 + (1 + u 2 )(1 + v 2 ) γ. As et(m) is a non-ecreasing function of δ, with limit γ as δ we first nee γ > that is our first conition. The secon conition gives the upper boun on δ to satisfy this inequality. 5 Numerical results The theoretical results inicate that solutions of the relaxation system (15) remain boune in L 2. Therefore, we expect that the instability obtaine for large initial ata or a small δ (through Turing mechanism) shoul lea to stiff graients. We present several numerical tests for the particular cubic system (25). They aim at showing that (i) the conitions of Lemma 4.3 are accurate an escribe the numerical transition to instability, (ii) stationary patterns are inee obtaine in this range of ata with stiff graients. These numerical results also show the variety of possible steay state, an interesting phenomena wiely stuie theoretically ([21] an the references therein). We have performe both 1D an 2D simulations in the following omains In interval = [, 1] (1D simulation) In rectangle = [, 2] [,.5] (in both cases = 1) In unit square =], 1[ 2 In 2D, the computations use an unstructure gri an a mixe finite element metho for space an backwar Euler scheme for time. The metho is alreay presente in [22]. We recall the eigenvectors of Laplace operator with Neumann bounary conition: for = [, 1], e n (x) = cos(nπx) an particularly, the first nonzero eigenvalue is π 2, associate to the eigenvector cos(π x). For = [, 2] [,.5], the eigenvectors are given by e n,m (x, y) = cos( nπx 2 ) cos(2mπy), the first nonzero eigenvalue is π 2 /4. We compare the theoretical formula of Lemma 4.3 an the numerical stability of the steay state. In all simulations we take u = 2, v = 1. In this case, instability might occur, since γ = 16 1 = 6 > an the limiting values of δ are given in the table 1. Therefore, in a first series of numerical tests, we choose the parameters δ 2 =.26 < δ 2 an δ2 =.27 > δ 2 in 1D an for δ2 =.1 < δ 2, an δ 2 =.11 > δ 2 in 2D. In both cases, we have obtaine relaxation to constant equilibrium when δ is taken larger than the critical value (for all the initial ata we have teste), an instability of the constant equilibrium when δ is smaller than the critical value. 13
Domain =], 1[ =], 2[ ],.5[ =], 1[ 2 Critical value δ 2.2649111.173644.2684 Table 1: Critical value of the parameter δ for Turing instability, compute from formula (26). (a) Intital conition (b) Steay state Figure 1: Initial conition (left) an steay state (right) in 2D simulations. The relaxation parameter δ 2 =.1 is small enough to fulfill conition (26). The scales for the solutions are not the same in the two figures. We illustrate the instability case with steay states in figure 3 for 1D simulations an in figure 1 for 2D simulations. For the 1D simulation, we took v 1 an u = 1.9 +.21I {].1,.6[}. Next we stuy the singularity that occurs on the transients for small relaxation parameter δ. Numerical solutions show that strong oscillations occur. In figure 2 we epict, for the same initial ata, the effect of δ on the solution at a given time. A Appenix: Michel Pierre s estimate Consier the problem { t u [a(t, x)u] =, u(t = ) = u, together with Neumann bounary conition in a boune omain. We enote Q T = (, T ) ). We assume that a(t, x) > is smooth an u is a weak solution. We can also assume withour lack of generality that u. Then we have the a priori estimate Lemma A.1 For any T >, we have a u L 2 (Q T ) C() u L 2 () + 2 u a L 2 (Q T ), (27) where C() is the constant of Poincaré Wirtinger s inequality. 14
(a) δ 2 = 2. 1 3 (b) δ 2 = 2. 1 4 (c) δ 2 = 5. 1 5 () δ 2 = 2. 1 5 Figure 2: Cuts, at a given time, in the y irection an in the mile of the omain = (, 1) 2 in 2D. The piecewise initial conition is also represente in ashe line. As expecte strong oscillations occur with species seggregation. These oscillations are stronger when δ is smaller. 15
(a) time t = (b) time t = 2 (c) time t = 5 () time t = 9 Figure 3: Time evolution for a 1D simulation for δ 2 =.25 < δ 2. This figure shows how a small perturbation is amplifie an creates a steay pattern. Because δ is large (close to δ ) there are not strong oscillations as in the case of smaller values. 16
Proof. Consier smooth functions F (t, x) an the solutions to the ajoint problem { t v + a(t, x) v = F (t, x), still with Neumann conitions. We have v(t = T ) =, uv = F u, t an thanks to the final conition for the ajoint problem, T u v = F u. (29) Multiplying (28) by v, we get integrating by parts on, we obtain, v 2 t 2 t v v + a v 2 = F v, + a v 2 ( F 2 2a + a 2 v 2 ), which gives after integration in time, using again v(t ) =, T v 2 + a v 2 an by consequence, T F 2 a, (28) v L 2 () F a L 2 (Q T ), (3) a v L 2 (Q T ) F a L 2 (Q T ). (31) We nee aitionally a boun on v that we erive as follows. We use again (28) to fin T T v ( a v F ) = a v F a + a, which gives, thanks to the Cauchy Schwarz inequality an (31), v 2 a L 2 (Q T ) F a L 2 (Q T ). (32) Finally, we get using Poincaré-Wirtinger ineqality, (32) an then (3), u v u (v v ) + u v C() u L 2 () v L 2 () + 2 u a L 2 (Q T ) F a L 2 (Q T ) C() u L 2 () F a L 2 (Q T ) + 2 u a L 2 (Q T ) F a L 2 (Q T ). 17
Back to (29), we conclue that T T ( F u = F au C() u a L 2 () + 2 u ) a L 2 (Q T ) F a L 2 (Q T ), which is equivalent to (27). B Energy for a particular cross-iffusion system A particular choice of cross-iffusion terms in (5) permits for an energy inequality even for negative secon orer matrices. This is the case of the system t U (U(1 + V 2 )) =, t V (V (1 + U 2 (33) )) =, still with Neumann bounary conitions an initial ata U, V. For this system, the energy is given by One can easily check that it hols E(x, t) := (1 + U 2 )(1 + V 2 ). t E(x, t) = 2U(1 + V 2 ) (U(1 + V 2 )) + 2(V (1 + U 2 )) (V (1 + U 2 )), which leas immeiately to (1 + U 2 )(1 + V 2 ) = 2 t (U(1 + V 2 )) 2 2 (V (1 + U 2 )) 2. It follows an a priori estimate in the space L t (L 2 x) that completes the L p tx boun prove in section 3. The system (33) is not always elliptic. This is relate to the non-convexity of this energy (still for large ata), an important ifference with the Shigezaa-Kawasaki prey-preator system which comes with a convex entropy functional ([6]). Acknowlegment This research has been supporte by INRIA-CONICYT scientific cooperation program. References [1] H. Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal., 12 (1988), pp. 895 919. [2], Dynamic theory of quasilinear parabolic equations. II. Reaction-iffusion systems, Differential Integral Equations, 3 (199), pp. 13 75. [3], Erratum: Dynamic theory of quasilinear parabolic systems. III. Global existence [Math. Z. 22 (1989), no. 2, 219 25; MR11386 (9i:35125)], Math. Z., 25 (199), p. 331. 18
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