Conservative cross diffusions and pattern formation through relaxation

Mostafa Bendahmane (1) , Thomas Lepoutre (2) , Americo Marrocco (2) , Benoît Perthame (3)

1 DIM - Departamento de Ingenieria Matematica
2 BANG - Nonlinear Analysis for Biology and Geophysical flows
3 LJLL - Laboratoire Jacques-Louis Lions

This paper is aimed at studying the formation of patches in a cross-diffusion system without reaction terms when the diffusion matrix can be negative but with positive self-diffusion. We prove existence results for small data and global a priori bounds in space-time Lebesgue spaces for a large class of 'diffusion' matrices. This result indicates that blow-up should occur on the gradient. One can tackle this issue using a relaxation system with global solutions and prove uniform a priori estimates. Our proofs are based on a duality argument à la M. Pierre which we extend to treat degeneracy and growth of the diffusion matrix. We also analyze the linearized instability of the relaxation system and a Turing type mechanism can occur. This gives the range of parameters and data for which instability can occur. Numerical simulations show that patterns arise indeed inthis range and the solutions tend to exhibit patches with stiff gradients on bounded solutions, in accordance with the theory.

Domaines: Analyse numérique [math.NA]

Mostafa Bendahmane, Thomas Lepoutre, Americo Marrocco, Benoît Perthame. Conservative cross diffusions and pattern formation through relaxation. 2009. ⟨inria-00368595⟩