Well-posedness of the Muskat problem withH2initial data

Abstract

We study the dynamics of the interface between two incompressible fluids in a twodimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small H 2 perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for H 2 initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.≤ C(|h 1 | 2 + |h 2 | 2 )|h| 1.75 + C(|h 1 | 1.75 + |h 2 | 1.75 + 1)|h| 2 , so we conclude that ∇Q 1.25,± + w 1.25,± ≤ C

Well‐Posedness and Global Behavior of the Peskin Problem of an Immersed Elastic Filament in Stokes Flow

“…For nearly flat interfaces, the linearization of the Muskat problem has the same symbol as the Peskin problem considered here, and one expects similar local well-posedness and stability results so long as condition jX j > 0 holds, see, for example, [1,7,9,10,13,40]. In the simplest setup in two dimensions, the Muskat problem features two fluids in porous media whose dynamics are governed by Darcy's law.…”

Growth in the Muskat problem

“…The previous result was given by Constantin, Córdoba, Gancedo & Strain [23] (see also [6,20]). Although the solution enjoys this decay of the relatively strong L ∞ norm and a energy balance that controls the velocity in L 2 (0, ∞; L 2 (R 2 )), this is not enough to obtain a global existence result of any kind.…”

A survey for the Muskat problem and a new estimate

“…Finally we would like to quote some recent articles where local-in-time existence is shown of classical solution for large and low regular initial data. For the one fluid case (µ 1 = ρ 1 = 0) see [30] and [17] for the density jump case. If µ 2 = µ 1 = µ and τ = 0, it is possible to obtain decay of the L ∞ norm of the interface for arbitrary initial data (see [22]).…”