Bounded weak solutions to a class of degenerate cross-diffusion systems
Annales Henri Lebesgue, Volume 6 (2023), pp. 847-874.


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


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 ( n ) n2 of Liapunov functionals such that  n is equivalent to the L n -norm for each n2 and n 1/n controls the L -norm in the limit n. 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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