BOUNDED WEAK SOLUTIONS TO THE THIN FILM MUSKAT PROBLEM VIA AN INFINITE FAMILY OF LIAPUNOV FUNCTIONALS

Philippe Laurençot (1) , Bogdan-Vasile Matioc

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 ≥ 2 we construct an homogeneous polynomial of degree n, which is convex on [0, ∞)^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.

Mots clés: degenerate parabolic system; cross-diffusion; boundedness; Liapunov functionals; global existence

Domaines: Equations aux dérivées partielles [math.AP]

Philippe Laurençot, Bogdan-Vasile Matioc. BOUNDED WEAK SOLUTIONS TO THE THIN FILM MUSKAT PROBLEM VIA AN INFINITE FAMILY OF LIAPUNOV FUNCTIONALS. Transactions of the American Mathematical Society, 2022, 375 (8), pp.5963--5986. ⟨hal-03362856⟩