Bounded weak solutions to a class of degenerate cross-diffusion systems

Laurençot, Philippe; Matioc, Bogdan-Vasile

doi:10.5802/ahl.179

Laurençot, Philippe; Matioc, Bogdan-Vasile

Bounded weak solutions to a class of degenerate cross-diffusion systems

Annales Henri Lebesgue, Volume 6 (2023), pp. 847-874.

Metadata

DOI10.5802/ahl.179

Keywords Degenerate parabolic system, cross-diffusion, boundedness, Liapunov functionals, global existence

Abstract

Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence ${(ℰ_{n})}_{n \geq 2}$ of Liapunov functionals such that $ℰ_{n}$ is equivalent to the $L_{n}$ -norm for each $n \geq 2$ and $ℰ_{n}^{1 / n}$ controls the $L_{\infty}$ -norm in the limit $n \to \infty$ . Weak solutions are built by a compactness approach, special care being needed in the construction of the approximation in order to preserve the availability of the above-mentioned Liapunov functionals.

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