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 sc (R d ) where sc = 1 + d 2 . Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces H s (R d ), s > sc. 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.The incompressible fluid velocity u ± in each region is governed by Darcy's law:and div x,y u ± = 0 in Ω ± t .(1.6) (ii) (The two-phase problem) Consider ρ − > ρ + > 0 and µ ± > 0. Assume that the upper and lower boundaries are either empty or graphs of Lipschitz functions that do not touch the interface. The two-phase Muskat problem is locally well posed in H s (R d ) in the sense that any initial data in H s (R d ) satisfying the Rayleigh-Taylor condition leads to a unique solution in C([0, T ]; H s (R d )) for some T > 0.The starting point of our analysis is the fact that the Muskat problem has a very simple reformulation in terms of the Dirichlet-Neumann map G (see the definition (2.2) below); most strikingly, in the case of one fluid, it is equivalent to ∂ t η + G(η)η = 0.(1.12) See Proposition 2.1. This makes it clear that • Any precise result on the continuity of the Dirichlet-Neumann map leads to direct application for the Muskat problem. This is especially relevant in view of the recent intensive work in the context of water-waves [2,3,5,28,43]. 2• The Muskat problem is the natural parabolic analog of the water-wave problem and as such is a useful toy-model to understand some of the outstanding challenges for the water-wave problem.The second point above applies to the study of possible splash singularities, see [12,13]. Another problematic is the question of optimal low-regularity well-posedness for quasilinear problems. This seems a rather formidable problem for water-waves since the mechanism of dispersion is harder to properly pin down in the quasilinear case (see [1,2,3,4,29,42,50,56,57]), but becomes much more tractable in the case of the Muskat problem due to its parabolicity. This is the question we consider here.The Muskat problem exists in many incarnations: with or without viscosity jump, with or without surface tension, with or without bottom, with or without permeability jump, in 2d or 3d, when the interface are graphs or curves. . . Our main objective is to provide a flexible approach that covers many aspects at the same time and provides almost sharp well-posedness results. The main questions t...

A truncated real interpolation method and characterizations of screened Sobolev spaces

“…They used this to characterize the solvability of certain quasilinear elliptic boundary value problems in these domains. This trace theory has also been used in recent studies of the Muskat problem by Nguyen and Pausader [35], Nguyen [34], and Flynn and Nguyen [18].…”

Global solutions for the Muskat problem in the scaling invariant Besov space $\dot B^1_{\infty, 1}$

“…The latter was then improved to the homogeneous space Ḣ1 (R) ∩ Ḣ 3 2 + (R) [1], which is natural since the PDE annihilates constants. For arbitrary viscosity contrast and in all dimensions, local well-posedness was obtained in [33] for H 1+ d 2 +ε (R d ) data. See also [1] for the one-phase case.…”

“…See also [1] for the one-phase case. The method developed in [33] could also handle the effect of surface tension [34] and the zero surface tension limit [23] at the same regularity level. The only existing scaling invariant (modulo low-frequency assumptions) existence and uniqueness results are [3,4] for the case of no viscosity contrast.…”

Mixing solutions for the Muskat problem with variable speed

“…The case ρ + < ρ − is called the stable regime. In the stable case, this equation is locally well-posed in H 3 (R), see [7,9], as well as [1,2,19,23] for recent developments. In the unstable case, however, which is our focus in this article, we have an ill-posed problem (see [9,25]), and there are no general existence results for (6) known.…”