The Muskat problem with surface tension and equal viscosities in subcritical L_p-Sobolev spaces

Anca-Voichita Matioc,Bogdan-Vasile Matioc

doi:10.1007/s41808-021-00104-1

Anca-Voichita Matioc, Bogdan-Vasile Matioc

Open Access

https://doi.org/10.1007/s41808-021-00104-1

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Abstract

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces W^s_p(mathbb {R}), where {pin (1,2]} and {sin (1+1/p,2)}. This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in W^{overline{s}-2}_p(mathbb {R}), where {overline{s}in (1+1/p,s)}. Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

Highlights

In this paper we study the evolution equation tf (t, x) = k 2 PV ∫R y+ y2 xf (t, x) [x,y]f (t) + [x,y]f (t) 2 x((f ) − Δ f )(t, x − y) dy,(1a) which is defined for t > 0 and x ∈ R
(f ) − Δ f )(t, x − y) dy, (1a) which is defined for t > 0 and x ∈ R
The problem (1a) describes the two-dimensional motion of two fluids with equal viscosities − = + = and general densities − and + in a vertical/horizontal homogeneous porous medium which is identified with R2

Summary

Introduction

(1a) which is defined for t > 0 and x ∈ R. Concerning the well-posedness of the Muskat problem with surface tension effects, this property has been investigated in bounded (layered) geometries in [14, 16, 17, 33, 36] where abstract parabolic theories have been employed in the analysis, the approach in [23] relies on Schauder’s fixed-point theorem, and in [10] the authors use Schauder’s fixed-point theorem in a setting which allows for a sharp corner point of the initial geometry. The results on the Muskat problem with surface tension in the unbounded geometry considered in this paper (and possibly in the general case of fluids with different viscosities) are more recent, cf [8, 24,25,26, 30, 31, 39]. This would require to establish real-analytic dependence of the right-hand side of (1a) on f in the functional analytic framework considered in Sect. 3, which is much more involved than showing the smooth dependence (see [25, Proposition 5.1] for a related proof of real-analyticity)

Notation

Outline

Some singular integral operators

A functional analytic framework for the Muskat problem