Some new results on triangular cross diffusion

Transcription

1 Some new results on triangular cross diffusion Laurent Desvillettes IMJ-PRG, Université Paris Diderot June 20, 2017

2 Results obtained in collaboration with Yong-Jung Kim, Changwook Yoon, KAIST, Daejeon, S. Korea Thomas Lepoutre, Univ. Lyon 1 and INRIA Ayman Moussa, Univ. P. & M. Curie Ariane Trescases, Cambridge Univ.

3 Reaction Cross diffusion systems considered here Unknowns: u := u(t, x) 0, v := v(t, x) 0, for t 0, x. Equations: ) t u x (u A(u, v) = R 1 (u, v), ) t v x (v B(u, v) = R 2 (u, v). where A, B 0, R 1 (0, v) 0, R 2 (u, 0) 0

4 Reaction Cross diffusion systems: triangular case Unknowns: u := u(t, x) 0, v := v(t, x) 0, for t 0, x. Equations: ) t u x (u A(u, v) = R 1 (u, v), ) t v x (v B 0 (v) = R 2 (u, v). where A, B 0, R 1 (0, v) 0, R 2 (u, 0) 0 triangular case: One assumes that B(u, v) := B 0 (v), or even B(u, v) := B 0

5 Reaction Cross diffusion systems: the non triangular SKT system (without reaction terms) Shigesada, Kawasaki, Teramoto, 1979 Equations: ) t u x (d 1 u + a 12 u v = 0, ) t v x (d 2 v + a 21 u v = 0. Interpretation: Due to competition, the individuals of each species increase their diffusion rate when the density of the other species increases.

6 Reaction Cross diffusion systems: the non triangular SKT system (without reaction terms) L. Chen, A. Jüngel, 2006 Equations: ) t u x (d 1 u + a 12 u v = 0, ) t v x (d 2 v + a 21 u v = 0. Lyapunov functional: H(u, v) = a 21 (u ln u u + 1) dx + a 12 (v ln v v + 1) dx (uses homogeneous Neumann BC).

7 Reaction Cross diffusion systems: the non triangular SKT system (without reaction terms) Sketch of proof of existence: One writes the entropy (Lyapunov functional) estimate under the form T H(u(T ), v(t )) + D(u(s), v(s)) ds = H(u(0), v(0)), 0 where D(u, v) = a 12 d 1 + a 12 a 21 x u 2 dx + a 21 d 2 u u v x u u + xv v 2 dx. x v 2 dx v

8 Reaction Cross diffusion systems: the non triangular SKT system (without reaction terms) Compactness in the x variable is extracted from the Fisher informations estimates T x u 2 T x v 2 dxdt C T, dxdt C T u v 0 Compactness in the t variable is obtained thanks to variants of Aubin-Lions lemma (interpolation between L 2 t (Hx 1 ) and Ht 1 (Hx 2 )), cf. Th. Lepoutre, A. Moussa Finally, an approximated system is used (hard!) Duality lemmas can be useful (especially if l.h.s, that is reaction terms, are considered). 0

9 Example of a recent result for more general non-triangular systems Theorem (LD, Lepoutre, Moussa - LD, Lepoutre, Moussa, Trescases) Let be a smooth open subset of R N We assume that β 12 > 0, β 21 > 0, β 12 β 21 < 1, and D i > 0, a 12 > 0, a 21 > 0. Let (u in, v in ) be initial data in L 2 (), then there exists a weak solution to eq. t u x (D 1 u + a 12 v β12 u) = 0, t v x (D 2 v + a 21 u β21 v) = 0, with Neumann boundary conditions on, and initial data (u in 1, uin 2 ). Main difficulty: Find a Lyapunov functional, (cf. Chen, Daus, LD, Jüngel, Lepoutre, A. Moussa). Case β 12 β 21 > 1?

10 First example of triangular system: triangular SKT model for competing populations Equations: ) t u x (D 1 u + a 12 u v = (r 1 a 1 u b 1 v) u, t v D 2 x v = (r 2 b 2 u a 2 v) v. Modeling assumption: The two species are in competition and diffuse in their environment. The individuals of the first species are the only one who increase their diffusion rate when the concentration of the other species increases.

11 Second example of triangular system: Model coming out of chemotoxis Equations: t u d 1 x (u v k ) = 0, t v d 2 x v = b u a v. The parameters are d i, b, a, k > 0. The concentration of chemoattractant is v and the number density of cells is u.

12 Remark: results of existence for the triangular system Lou, Ni, Wu: Existence of solutions for triangular SKT systems in dimension 1, 2; Choi, Lui, Yamada: Existence of solutions for triangular SKT systems in dimension 5; LD, Trescases: Existence for a large class of (non necessarily quadratic) triangular systems including SKT and the system coming out of chemotaxis; independent of dimension.

13 Main a priori estimates Usually, it is easy to prove that u L ([0, T ]; L 1 ()) by integrating the equation for u, but this is not a good space to start with when one wants to use the properties of the heat equation. Instead one writes down a Duality Lemma: We consider an open smooth (C 2 ) and bounded subset of R N, and T > 0. We also consider C > 0 a constant, and M := M(t, x) a smooth function. Let u 0 be a strong solution of the inequality t u(t, x) x (M(t, x) u(t, x)) C, satisfying Neumann boundary condition. Then T 0 T ) u 2 M 4 ( C T + u(0, ) L 2 ()) (C(, 2 T ) + M. 0

14 If Assumption: u(t, x) 0, 0 < c 1 M(t, x) c 2. multiplying by x u, we get u H 2. t u(t, x) M(t, x) x u(t, x) C, multiplying by u, we get u H 1. t u(t, x) x (M(t, x) x u(t, x)) C, t u(t, x) x (M(t, x) u(t, x)) C, multiplying by the solution of the dual problem, we get u L 2.

15 Consequence of the duality lemma For the triangular SKT model, the maximum principle yields v L ([0,T ] ) C T. Then the duality lemma ensures that T 0 T ] u 2 (D 1 + a 12 v) C T [1 + (D 1 + a 12 v), 0 so that u L2 ([0,T ] ) C T. As a consequence, the properties of the heat kernel lead to the estimate t v L 2 ([0,T ] ) + xi x j v L 2 ([0,T ] ) C T ; xi v L 4 ([0,T ] ) C T.

16 Multiplicator method For the triangular SKT model, computing the derivative (in time) of u ln u u + 1 and integrating on yields t (u ln u u + 1) = ln u x (D 1 u + a 12 u v) But = (D 1 + a 12 v) xu 2 a 12 u x u x v. x u 2 x u x v ε + C ε u x v 2 u x u 2 ε + C ε u 2 + C ε x v 4. u

17 Proof of existence for the triangular SKT system Compactness in the x variable is extracted from the Fisher information estimate T x u 2 C T, u and the properties of the heat kernel for v. 0 Compactness in the t variable is obtained thanks to variants of Aubin-Lions lemma (interpolation between L 2 t (Hx 1 ) and Ht 1 (Hx 2 )), cf. Th. Lepoutre, A. Moussa Uniform integrability of u 2 is obtained thanks to a variant of the duality lemma Finally, an approximated system is used (easy!)

18 Conclusion For triangular reaction cross diffusion systems, duality lemmas are combined with classical multiplicator/heat kernel/maximum principle estimates. Improvements may come from better duality lemmas. For non triangular reaction cross diffusion systems, the main ingredient is Lyapunov functionals (or variants). The search for such functionals is very active. Approximated systems lead to many technical difficulties.