On the Cauchy problem for the Muskat equation with non-Lipschitz initial data

Abstract

This article is devoted to the study of the Cauchy problem for the Muskat equation. We consider initial data belonging to the critical Sobolev space of functions with three-half derivative in $L^2$, up to a fractional logarithmic correction. As a corollary, we obtain the first local and global well-posedness results for initial free surface which are not Lipschitz.

41 pages

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Cited by in corpus (4)

• Regularity of solutions to the Muskat equation
• Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data
• Entropy solutions to macroscopic IPM
• The Second Iterate of the Muskat Equation in Supercritical Spaces