On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data

Deng, Fan; Lei, Zhen; Lin, Fanghua

doi:10.1002/cpa.21669

Cited by 43 publications

(33 citation statements)

References 29 publications

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“…We note in particular that [25] allows for interfaces with large slope and [15,35] allow for viscosity jump. Global weak solutions were obtained in [17,26]. 3 As noted earlier there is a significant difference between small and large data for this quasilinear problem.…”

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confidence: 66%

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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confidence: 66%

A Paradifferential Approach for Well-Posedness of the Muskat Problem

Nguyen

Pausader

2020

Arch Rational Mech Anal

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“…They have proved that there are solutions such that at time t = 0 the interface is a graph, at a subsequent time t 1 > 0 the interface is not a graph and then at a later time t 2 > t 1 , the interface is C 3 but not C 4 . This result explains why it is interesting to prove the existence of solutions whose slopes can be arbitrarily large (as in [33,15,31,36]) or even infinite (as we proved in [5,6,4]).…”

Section: Introductionmentioning

confidence: 58%

“…This problem was extensively studied in the last decade. In particular, there are many local well-posedness results in sub-critical spaces: we refer to the works of Constantin, Gancedo, Shvydkoy and Vicol [25] for initial data in the Sobolev space W 2,p (R) for some p > 1, Cheng, Granero-Belinchón and Shkoller [22] and Matioc [38,39] for initial data in H s (R) with s > 3/2 (see also [3,41]), and Deng, Lei and Lin [33] for 2D initial data whose derivatives are Hölder continuous. The Muskat equation being parabolic, the proof of the local well-posedness results also provides global well-posedness results under a smallness assumption.…”

Section: Cauchy Problemmentioning

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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“…However, many of the mathematical studies on this topic are quite recent and they cover various physical scenarios and mathematical aspects related to the original model proposed in [52], cf. [6,7,9,[11][12][13][14][15][16][17][18][21][22][23][24][25]27,30,32,36,[38][39][40][41][42][43]48,49,49,53,54,58,[60][61][62] (see also [55,56] for some recent research on the compressible analogue of the Muskat problem, the so-called Verigin problem).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Well-Posedness and Stability Results for Some Periodic Muskat Problems

Matioc

2020

J. Math. Fluid Mech.

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We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space $$H^r({\mathbb {S}})$$ H r ( S ) for each $$r\in (2,3)$$ r ∈ ( 2 , 3 ) . When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh–Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of $$H^2({\mathbb {S}})$$ H 2 ( S ) defined by the Rayleigh–Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.

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On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data

Cited by 43 publications

References 29 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

Well-Posedness and Stability Results for Some Periodic Muskat Problems

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