Local solvability and turning for the inhomogeneous Muskat problem

Berselli, Luigi C.; Córdoba, Diego; Granero-Belinchón, Rafael

doi:10.4171/ifb/317

Cited by 33 publications

(56 citation statements)

References 28 publications

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“…Actually, as noted in [2], (at the time) it is(was) an open problem as to whether the problem is well-posed for initial data with a cusp. Theorem 3.1 was proved by Córdoba & Gancedo [32] using energy methods (a similar result for the case where the spatial domain is a strip, for the case of a porous medium with two different permeabilities and for the case of three fluids was proved by Córdoba, Granero-Belinchón and Orive [34] and Berselli, Córdoba and Granero-Belinchón [6] and Córdoba & Gancedo [36], respectively). A different proof (using a formulation for (1.6) based on the tangent angle and arclength) was given by Ambrose [2,1] (see also [96]).…”

Section: Well-posednessmentioning

confidence: 60%

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Growth in the Muskat problem

Granero-Belinchón

Lazar

2020

Math. Model. Nat. Phenom.

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Section: Well-posednessmentioning

confidence: 60%

“…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.…”

Section: Well-posednessmentioning

confidence: 61%

Growth in the Muskat problem

Granero-Belinchón

Lazar

2020

Math. Model. Nat. Phenom.

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“…The main questions that we do not address here are• The case of surface tension or jump in permeability (see e.g. [46,47,9,37,51]). This can also be covered by the paradifferential formalism, but we decided to leave it for another work in order to highlight the centrality of the Dirichlet-Neumann operator.…”

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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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“…Interesting scenarios consider 3D fluids, multiphase flows, boundary effects or permeability discontinuities (see for example [5,43]), etc. Different methods interact, raging from analytic to computerassisted proofs [38].…”

Section: Mathematical Resultsmentioning

confidence: 99%

“…The framework picked in this presentation allows us to reduce the problem from its original Eulerian variables formulation (eqs. (1,2,3,4,5)) to the self-evolution of an interface, hence the name contour evolution equation. It provides a simple way to linearize the system of equations to illustrate in a non-technical manner what is going on at the nonlinear level.…”

Section: Contour Evolution Equationmentioning

confidence: 99%

A survey for the Muskat problem and a new estimate

Gancedo

2016

SeMA

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This paper shows a summary of mathematical results about the Muskat problem. The main concern is well-posed scenarios which include the possible formation of singularities in finite time or existence of solutions for all time. These questions are important in mathematical physics but also have a strong mathematical interest. Stressing some recent results of the author, we also give a new estimate for the problem in the last section. Initial data with L 2 decay and slope less than one provide weak solutions which satisfy a parabolic inequality as in the linear regime.

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Local solvability and turning for the inhomogeneous Muskat problem

Cited by 33 publications

References 28 publications

Growth in the Muskat problem

Growth in the Muskat problem

A Paradifferential Approach for Well-Posedness of the Muskat Problem

A survey for the Muskat problem and a new estimate

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