Global well-posedness for the two-dimensional Muskat problem with slope less than 1

Cameron, Stephen

doi:10.2140/apde.2019.12.997

Cited by 60 publications

(48 citation statements)

References 23 publications

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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: 59%

“…In [26], Deng, Lei and Lin proved the existence of global in time solutions with large slopes, assuming some monotonicity assumption on the data. In [8], Cameron was able to prove a global existence result assuming that some critical quantity, namely the product of the maximal and minimal slopes, is smaller than 1. His result allows to consider 2 arbitrary large slopes.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 59%

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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“…The nonlinearity in the Muskat equation is more complicated. However, Cameron has succeeded in [9] to apply the method introduced by Kiselev-Nazarov-Volberg to prove the existence of global solutions in time when the product of the maximum and minimum slopes is less than 1 (see also [10,11]). Recently, many works have extended this last result.…”

Section: Introductionmentioning

confidence: 99%

On the dynamics of the roots of polynomials under differentiation

Alazard¹,

Lazar²,

Nguyen³

2021

Preprint

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This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence Tao on free convolution. Rafael Granero-Belinchón obtained a global well-posedness result for positive initial data small enough in a Wiener space, and recently Alexander Kiselev and Changhui Tan proved a global well-posedness result for any positive initial data in the Sobolev space H s (S) with s > 3/2. In this paper, we consider the Cauchy problem in the critical space H 1/2 (S). Two interesting new features, at this level of regularity, are that the equation can be written in the form ∂tu + V ∂xu + γΛu = 0, where γ is non-negative but not bounded from below and V / √ γ is not bounded. Therefore, the equation is only weakly parabolic. We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times. Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any positive initial data in H 1/2 (S). Our proof relies on sharp commutator estimates and introduces a strategy to prove a local well-posedness result in a situation where the lifespan depends on the profile of the initial data and not only on its norm.

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“…There are many possible ways to study the Hele-Shaw equation: to mention a few approaches we quote various PDE methods based on L 2 -energy estimates (see the works of Chen [15], Córdoba, Córdoba and Gancedo [24], Knüpfer and Masmoudi [36], Günther and Prokert [33], Cheng, Granero-Belinchón and Shkoller [16]), there are also methods based on functional analysis tools and maximal estimates (see Escher and Simonett [30], the results reviewed in the book by Prüss and Simonett [42] and Matioc [38,39]) or methods using harmonic analysis tools and contour integrals (see the numerous results reviewed in the survey papers by Gancedo [31] or Granero-Belinchón and Lazar [32]). For the related Muskat equation (a two-phase Hele-Shaw problem), maximum principles have played a key role to study the Cauchy problem, see [12,18,10,26] following the pioneering work of Constantin, Córdoba, Gancedo, Rodríguez-Piazza and Strain [17]. Such maximum principles have been obtained for general viscosity solutions of the Hele-Shaw equation by Kim [35], see also the recent work of Chang-Lara, Guillen and Schwab [14].…”

Section: Introductionmentioning

confidence: 99%

Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation

Alazard

Meunier

Smets

2020

Commun. Math. Phys.

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This article is devoted to the study of the Hele-Shaw equation. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancellations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equations in a natural way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapunov functions of different natures. We also deduce from these identities and previous works on the water wave problem a simple proof of the well-posedness of the Cauchy problem. The analysis contains two side results of independent interest. Firstly, we give a principle to derive estimates for the modulus of continuity of a PDE under general assumptions on the flow. Secondly we prove and give applications of a convexity inequality for the Dirichlet to Neumann operator.

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Global well-posedness for the two-dimensional Muskat problem with slope less than 1

Cited by 60 publications

References 23 publications

Paralinearization of the Muskat Equation and Application to the Cauchy Problem

Paralinearization of the Muskat Equation and Application to the Cauchy Problem

On the dynamics of the roots of polynomials under differentiation

Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation

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