Paralinearization of the Muskat Equation and Application to the Cauchy Problem

By Thomas Alazard, Omar Lazar. Archive for Rational Mechanics and Analysis, 2019

63 citations recorded by the source corpus.

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 $$\dot{H}^1(\mathbb {R})\cap \dot{H}^s(\mathbb {R})$$ H ˙ 1 ( R ) ∩ H ˙ s ( R ) with $$s>3/2$$ s > 3 / 2 . This paper is essentially self-contained and does not rely on general results from paradifferential calculus.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a22cacf9fa51205e5de45a0d17afafe6e4f14ec6

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In Applied Mathematics

1. Quoc-Hung Nguyen
2. Thomas Alazard

In Control and Systems Engineering

1. Quoc-Hung Nguyen
2. Thomas Alazard

In Mathematical Physics

1. Quoc-Hung Nguyen
2. Thomas Alazard

Citations in the corpus

• Endpoint Sobolev Theory for the Muskat Equation (2020, methodology): “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 [2,24,28,36,44,46]).”