Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionals

Laurençot, Philippe; Matioc, Bogdan-Vasile

doi:10.1090/tran/8688

Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionals
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by Philippe Laurençot and Bogdan-Vasile Matioc PDF

Trans. Amer. Math. Soc. 375 (2022), 5963-5986 Request permission

Abstract:

A countably infinite family of Liapunov functionals is constructed for the thin film Muskat problem, which is a second-order degenerate parabolic system featuring cross-diffusion. More precisely, for each $n\geq 2$ we construct an homogeneous polynomial of degree $n$, which is convex on $[0,\infty )^2$, with the property that its integral is a Liapunov functional for the problem. Existence of global bounded non-negative weak solutions is then shown in one space dimension.

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