11 Citations
Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations
- Mathematics
- 2020
We establish that the $C^{1,\gamma}$ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of…
Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation
- MathematicsCommunications in Mathematical Physics
- 2020
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…
Convexity and the Hele–Shaw Equation
- MathematicsWater Waves
- 2020
Walter Craig’s seminal works on the water-wave problem established the importance of several exact identities: Zakharov’s hamiltonian formulation, shape derivative formula for the…
Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics
- Mathematics
- 2020
This paper is motivated by the study of Lyapunov functionals for four equations describing free surface flows in fluid dynamics: the Hele-Shaw and Mullins-Sekerka equations together with their…
Global well-posedness for the one-phase Muskat problem
- Mathematics
- 2021
The free boundary problem for a two-dimensional fluid filtered in porous media is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw…
A Paradifferential Approach for Well-Posedness of the Muskat Problem
- MathematicsArchive for Rational Mechanics and Analysis
- 2020
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…
Coercivity of the Dirichlet-to-Neumann operator and applications to the Muskat problem
- Mathematics
- 2022
A BSTRACT . We consider the Dirichlet-to-Neumann operator in strip-like and half-space domains with Lipschitz boundary. It is shown that the quadratic form generated by the Dirichlet-to-Neumann…
Refined Rellich boundary inequalities for the derivatives of a harmonic function
- Mathematics, Philosophy
- 2022
. The classical Rellich inequalities imply that the L 2 -norms of the normal and tangential derivatives of a harmonic function are equivalent. In this note, we prove several refined inequalities,…
Regularity of the free boundary for the two-phase Bernoulli problem
- MathematicsInventiones mathematicae
- 2021
We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we…
References
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Regularity of the free boundary in parabolic phase-transition problems
- Mathematics
- 1996
In this paper we start the study of the regularity properties of the free boundary, for parabolic two-phase free boundary problems. May be the best known example of a parabolic two-phase free…
From the free boundary condition for Hele-Shaw to a fractional parabolic equation
- Mathematics
- 2016
We propose a method to determine the smoothness of sufficiently flat solutions of one phase Hele-Shaw problems. The novelty is the observation that under a flatness assumption the free boundary…
On the regularity theory of fully nonlinear parabolic equations
- Mathematics
- 1990
Recently M. Crandall and P. L. Lions [3] developed a very successful method for proving the existence of solutions of nonlinear second-order partial differential equations. Their method, called the…
On the regularity theory of fully nonlinear parabolic equations: II
- Mathematics
- 1992
Recently M. Crandall and P. L. Lions [3] developed a very successful method for proving the existence of solutions of nonlinear second-order partial differential equations. Their method, called the…
Min–max formulas for nonlocal elliptic operators
- MathematicsCalculus of Variations and Partial Differential Equations
- 2019
In this work, we give a characterization of Lipschitz operators on spaces of $C^2(M)$ functions (also $C^{1,1}$, $C^{1,\gamma}$, $C^1$, $C^\gamma$) that obey the global comparison property-- i.e.…
Regularity of Free Boundaries in Obstacle-type Problems
- Mathematics
- 2012
The regularity theory of free boundaries flourished during the late 1970s and early 1980s and had a major impact in several areas of mathematics, mathematical physics, and industrial mathematics, as…
Neumann Homogenization via Integro-Differential Operators
- Mathematics
- 2014
In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic…
On the global existence for the Muskat problem
- Mathematics, Computer Science
- 2013
This work proves an L2(R) maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface, and takes advantage of the fact that the bound ‖∂xf0‖L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.
A free-boundary problem for the heat equation arising in flame propagation
- Mathematics
- 1995
We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation…
On the existence of convex classical solutions to a generalized Prandtl-Batchelor free-boundary problem-II
- Mathematics
- 2002
Abstract. We give an analytical proof of the existence of convex classical solutions for the (convex) Prandtl-Batchelor free boundary problem in fluid dynamics. In this problem, a convex vortex core…