Bounded Weak Solutions to a Class of Degenerate Cross-Diffusion Systems

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 $({\mathcal{E}}_n{)}_{n\ge 2}$ of Liapunov functionals such that ${\mathcal{E}}_n$ is equivalent to the $L_n$-norm for each $n\ge 2$ and ${\mathcal{E}}_n^{1\mathrm{/}n}$ controls the $L_{\mathrm{\infty }}$-norm in the limit $n\to \mathrm{\infty }$. 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.