Uniqueness of strong solutions and weak-strong stability in a system of cross-diffusion equations

Judith Berendsen (1) , Martin Burger (2) , Virginie Ehrlacher (3, 4) , Jan-Frederik Pietschmann (1)

1 Chemnitz University of Technology / Technische Universität Chemnitz
2 FAU - Friedrich-Alexander Universität Erlangen-Nürnberg = University of Erlangen-Nuremberg
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 MATHERIALS - MATHematics for MatERIALS

Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in [Ehrlacher2017]. Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence of strong solutions. The proof relies on the use of an appropriate approximation and a fixed-point argument. In addition, a weak-strong stability result is obtained for this system in dimension one which implies uniqueness.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Judith Berendsen, Martin Burger, Virginie Ehrlacher, Jan-Frederik Pietschmann. Uniqueness of strong solutions and weak-strong stability in a system of cross-diffusion equations. Journal of Evolution Equations, 2019, ⟨10.1007/s00028-019-00534-4⟩. ⟨hal-01967899⟩