A gradient flow approach to a thin film approximation of the Muskat problem

Philippe Laurencot (1) , Bogdan-Vasile Matioc (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Fakultät für Mathematik [Wien]

A fully coupled system of two second-order parabolic degenerate equations arising as a thin film approximation to the Muskat problem is interpreted as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of weak solutions. The availability of two Liapunov functionals turns out to be a central tool to obtain the needed regularity to identify the Euler-Lagrange equation in the variational scheme.

Mots clés: thin film; degenerate parabolic system; gradient flow; Wasserstein distance

Domaines: Equations aux dérivées partielles [math.AP]

Philippe Laurencot, Bogdan-Vasile Matioc. A gradient flow approach to a thin film approximation of the Muskat problem. Calculus of Variations and Partial Differential Equations, 2013, 47, pp.319-341. ⟨10.1007/s00526-012-0520-5⟩. ⟨hal-00636647⟩