A paradifferential approach for well-posedness of the Muskat problem

Abstract

We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension $d$ of the interface. The Muskat problem is scaling invariant in the Sobolev space $H^{s_c}({\mathbb{R}}^d)$ where $s_c=1+\frac{d}{2}$. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces $H^s({\mathbb{R}}^d)$, $s>s_c$. Moreover, the rigid boundaries are only required to be Lipschitz and can have arbitrarily large variation. The Rayleigh-Taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid. The starting point of this work is a reformulation solely in terms of the Drichlet-Neumann operator. The key elements of proofs are new paralinearization and contraction results for the Drichlet-Neumann operator in rough domains.

Contraction estimates for the Dirichlet-Neumann operator extended to the natural regularity range. Some proof details added and references updated

References in corpus (4)

• A maximum principle for the Muskat problem for fluids with different densities
• Paralinearization of the Muskat equation and application to the Cauchy problem
• Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation
• The vanishing surface tension limit of the Muskat problem

Cited by in corpus (5)

• Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
• The vanishing surface tension limit of the Muskat problem
• 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