On the global existence for the Muskat problem

Abstract

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L 2 (R) maximum principle, in the form of a new "log" conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy f 0 L ∞ < ∞ and ∂ x f 0 L ∞ < 1. We take advantage of the fact that the bound ∂ x f 0 L ∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance f 1 ≤ 1/5. Previous results of this sort used a small constant ǫ ≪ 1 which was not explicit [5,16,7,12].

Global Stability and Decay for the Classical Stefan Problem

“…Their methods rely on a boundary-integral formulation for the Muskat problem, together with the Taylor sign condition. In a subsequent work [12], various global existence results were established. An overview can be found in [10].…”

A survey for the Muskat problem and a new estimate

“…For gravity and surface tension interaction with boundary values see [34]. Those global existence results have been extended in some situations assuming initial smallness for critical norms with respect to the scaling [16,39], and showing instant analyticity in [7]. In works [16,15] some results of global in time regularity of classical solutions are shown with µ 1 = µ 2 , τ = 0 and medium-size initial slope in the Wiener algebra, i.e ∫ |ξ||f (ξ)|dξ ≤ c 0 with c 0 an explicit constant.…”

“…holds [16], which does not give a chance of gaining any regularity at the level of f . This can be easily shown by the bound ∫…”

“…The convergence as ε → 0 + is, up to a subsequence, strong in L ∞ on compact sets of [0, T ] × R and weak-start in L ∞ (0, T ; W 1,∞ ). Furthermore, it is possible to find (see discussion in [16] above Remark 4.4 for more details)…”

On the generalized Buckley-Leverett equation

“…By assumption 0 ≤ u(x, 0) ≤ 1. We apply the same technique as Córdoba & Córdoba [11] (see also [7,13,8 …”