Growth in the Muskat problem

Granero-Belinchón, Rafael; Lazar, Omar

doi:10.1051/mmnp/2019021

Cited by 21 publications

(22 citation statements)

References 92 publications

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“…Interestingly, the Muskat problem is mathematically analogous to the Hele-Shaw problem [39,40] for viscous flows between two closely spaced parallel plates. We will mostly discuss the issue of well-posedness and refer to [12,13,33] for interesting results on singularity formation and to [36,32] for recent reviews on the Muskat problem. In the case of small data and infinite depth, global strong solutions have been constructed in subcritical spaces [11,14,15,16,17,18,19,22,25,54] and in critical spaces [35].…”

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confidence: 99%

A Paradifferential Approach for Well-Posedness of the Muskat Problem

Nguyen

Pausader

2020

Arch Rational Mech Anal

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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...

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A Paradifferential Approach for Well-Posedness of the Muskat Problem

Nguyen

Pausader

2020

Arch Rational Mech Anal

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“…1 In [21], Córdoba and Gancedo discovered a formulation of the previous system based on contour integral, which applies whether the interface is a graph or not. The latter work opened the door to the solution of many important problems concerning the Cauchy problem or blow-up solutions (see [19,9,10,11], more references are given below as well as in the survey papers [30,31]). This formulation is a compact equation where the unknown is the parametrization of the free surface, namely a function f = f (t, x) depending on time t ∈ R + and x ∈ R, satisfying…”

Section: Introductionmentioning

confidence: 99%

“…In [14], Cheng, Granero-Belinchón, Shkoller proved the well-posedness of the Cauchy problem in H 2 (R) (introducing a Lagrangian point of view which can be used in a broad setting, see [34]) and Constantin, Gancedo, Shvydkoy and Vicol ( [18]) considered rough initial data which are in W 2,p (R) for some p > 1, as well they obtained a regularity criteria for the Muskat problem. We refer also to the recent work [31] where a regularity criteria is obtained in terms of a control of some critical quantities. Many recent results are motivated by the fact that, loosely speaking, the Muskat equation has to do with the slope more than with the curvature of the fluid interface.…”

Section: Introductionmentioning

confidence: 99%

Paralinearization of the Muskat Equation and Application to the Cauchy Problem

Alazard

Lazar

2020

Arch Rational Mech Anal

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We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spacesḢ 1 (R) ∩Ḣ s (R) with s > 3/2. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.

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“…This strategy proved to be successful for various versions of the Muskat (or twophase Hele-Shaw) problem, see e.g. the survey articles [6,7]. While the constant coefficient elliptic operator underlying the Muskat problem is simply the Laplacian, the related moving boundary problems of quasistationary Stokes flow are based on the Stokes operator (which is also elliptic in a sense that can be made precise).…”

Section: Introductionmentioning

confidence: 99%

Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

Matioc

Prokert²

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

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Growth in the Muskat problem

Cited by 21 publications

References 92 publications

A Paradifferential Approach for Well-Posedness of the Muskat Problem

A Paradifferential Approach for Well-Posedness of the Muskat Problem

Paralinearization of the Muskat Equation and Application to the Cauchy Problem

Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

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