Paralinearization of the Muskat equation and application to the Cauchy problem

Abstract

We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spaces ${\dot{H}}^1(\mathbb{R})\cap {\dot{H}}^s(\mathbb{R})$ with $s>3/2$. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.

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Cited by in corpus (8)

• A paradifferential approach for well-posedness of the Muskat problem
• Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation
• Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
• The vanishing surface tension limit of the Muskat problem
• 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