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 $\mathcal{E}_n^{1/n}$ 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.