A Paradifferential Approach for Well-Posedness of the Muskat Problem

By Huy Quang Nguyen, Benoit Pausader. Archive for Rational Mechanics and Analysis, 2019

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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 $$H^{s_c}({\mathbb {R}}^d)$$ H s c ( R d ) where $$s_c=1+\frac{d}{2}$$ s c = 1 + d 2 . Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces $$H^s({\mathbb {R}}^d)$$ H s ( R d ) , $$s>s_c$$ 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e9eaf910e9557ef97ac1db627ce0e31ba4a164c1

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Endpoint Sobolev Theory for the Muskat Equation (2020, background): “There are now many different proofs that the Cauchy problem is well-posed, locally in time, on the sub-critical Sobolev spaces Hs(R) with s > 3/2 (see [21, 25, 33, 41, 2, 43]).”