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 of Liapunov functionals such that is equivalent to the -norm for each and controls the -norm in the limit . 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.
[ELM11] Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 4, pp. 583-598 | DOI | Numdam | MR | Zbl