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 …

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 …

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 …

The free boundary problem for a two‐dimensional fluid permeating a porous medium is
studied. This is known as the one‐phase Muskat problem and is mathematically equivalent …

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We prove that the Cauchy problem is well-posed in a strong sense and in a general setting.
Our main result is the construction of an abstract semi-flow for the Hele-Shaw problem within …

This paper is motivated by the study of Lyapunov functionals for the Hele-Shaw and Mullins-
Sekerka equations describing free surface flows in fluid dynamics. We prove that the L2 …

We study the regularity of the interface between the disjoint supports of a pair of nonnegative
subharmonic functions. The portion of the interface where the Alt–Caffarelli–Friedman (ACF) …

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 …

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 …