Global well-posedness for the 2D stable Muskat problem in H2/3

Authors: Diego Córdoba (ICMAT-CSIC) and Omar Lazar

Source: Annales scientifiques de L’ Ecole normale supérieure

Date of publication: 2021



The Muskat problem models the motion of an interface separating two incompressible fluids in a porous medium. Let’s assume that both fluids are immiscible. The non-mixture condition allows one to consider the interface between these two fluids. Darcy’s law, together with the conservation of mass and the incompressibility of the fluids is the so-called incompressible porous media system (IPM system).

For the Muskat problem the IPM system can be rewritten in terms of the dynamics of the interface in between both fluids by using standard potential theory and this leads to the evolution equation of the interface (Muskat equation). Recently, it has been proved that some solutions can pass from the stable to the unstable regime and return back to the stable regime before the solution breaks down. This shift of stability phenomena illustrates the unpredictability of the solutions to the Muskat equation even starting in the stable regime. Moreover, there is numerical evidence of initial data whose slope is 50 that develops an infinite slope in finite time.

In this paper, the authors develop H2/3 critical theory under an arbitrary bounded slope assumption. The approach is completely new and is based on a reformulation of the usual Muskat equation. This new formulation allows one to take advantage of the oscillations which are crucial in this problem. There are many ways of measuring smoothness while trying to do a priori estimates in critical spaces. Contrarily to (almost) all previous works in the Muskat equation we shall never split the study into high/low frequencies or small/big increment in the finite difference operator. On the contrary, we shall consider the interaction between both and the Besov spaces techniques will be the main tool to achieve this. It is worth saying that the new formulation of the problem turns out to be crucial to prove the main theorems of this paper since it gives new features that are very difficult to see in the original formulation.

This article is a significant step in understanding the theory of global well-posedness of large solutions in the Lipschitz space. Indeed, the main result of this paper is the global well-posedness of strong solution in H5/2H3/2 under a smallness assumption on the norm of the initial data. It would not be possible to prove a global result for all data in the Lipschitz space since there has been shown that there are solutions with initial data having a (relatively) high slopes that become singular in finite time showing the instability of the Cauchy problem associated to initial data in critical spaces.