Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem

On the Cauchy problem for the Muskat equation. II: Critical initial data

We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in $L^2$. Moreover, we prove that the solution exists globally in time under a smallness assumption.

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

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.

Paralinearization of the Muskat equation and application to the Cauchy problem

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.

The Muskat problem with $C^1$ data

In this paper we prove that the Cauchy problem of the Muskat equation is wellposed locally in time for any initial data in $\dot C^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$.

带表面张力的Muskat问题任意大Lipschitz解的全局适定性

Global well-posedness of arbitrarily large Lipschitz solutions for the Muskat problem with surface tension

该研究针对带表面张力的二维Muskat问题，证明了满足临界空间小、Lipschitz范数可大的正则初值对应的柯西问题存在唯一全局解，允许界面斜率任意大，推进了该问题的全局适定性理论。

Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime

We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration in [CCF16]. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.