Thomas Alazard
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Global well-posedness of a 2D fluid-structure interaction problem with free surface Open
This paper is devoted to the analysis of the incompressible Euler equation in a time-dependent fluid domain, whose interface evolution is governed by the law of linear elasticity. Our main result asserts that the Cauchy problem is globally…
Paracomposition Operators and Paradifferential Reducibility Open
Reducibility methods, aiming to simplify systems by conjugating them to those with constant coefficients, are crucial for studying the existence of quasiperiodic solutions. In KAM theory for PDEs, these methods help address the invertibili…
Nonlinear interpolation and the flow map for quasilinear equations Open
We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those int…
Paralinearization of free boundary problems in fluid dynamics Open
A classical topic in the mathematical theory of hydrodynamics is to study the evolution of the free surface separating air from an incompressible perfect fluid. The goal of this survey is to examine this problem for two important sets of e…
The Hele-Shaw semi-flow Open
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 general fluid domains (enabling, for instance, changes…
KAM via Standard Fixed Point Theorems Open
With a mere usage of well-established properties of para-differential operators, the conjugacy equations in several model KAM problems are converted to para-homological equations solvable by standard fixed point argument. Such discovery gr…
Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics Open
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 L^2 -norm of the free surface elevation and the area of the fre…
Damping for fractional wave equations and applications to water waves Open
Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give …
Virial theorems and equipartition of energy for water-waves Open
We study several different aspects of the energy equipartition principle for water waves. We prove a virial identity that implies that the potential energy is equal, on average, to a modified version of the kinetic energy. This is an exact…
Refined Rellich boundary inequalities for the derivatives of a harmonic function Open
The classical Rellich inequalities imply that the -norms of the normal and tangential derivatives of a harmonic function are equivalent. In this note, we prove several refined inequalities, which make sense even if the domain is not Lipsch…
A Morawetz inequality for water waves Open
We consider gravity water waves in two space dimensions, with finite or\ninfinite depth. Assuming some uniform scale invariant Sobolev bounds for the\nsolutions, we prove local energy decay (Morawetz) estimates globally in time.\nOur resul…
Refined Rellich boundary inequalities for the derivatives of a harmonic function Open
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, which make sense even if the domain is not L…
Traveling wave solution for a coupled incompressible Darcy's free\n boundary problem with surface tension Open
We study an incompressible Darcy's free boundary problem, recently introduced\nin [22]. Our goal is to prove the existence of non-trivial traveling wave\nsolutions and thus validate the interest of this model to describe cell\nmotility. Th…
Traveling wave solution for a coupled incompressible Darcy's free boundary problem with surface tension Open
We study an incompressible Darcy's free boundary problem, recently introduced in [22]. Our goal is to prove the existence of non-trivial traveling wave solutions and thus validate the interest of this model to describe cell motility. The m…
Cauchy Theory for the Water Waves System in an Analytic Framework Open
In this paper we consider the Cauchy problem for gravity water waves, in a\ndomain with a flat bottom and in arbitrary space dimension. We prove that if\nthe data are of size $\\varepsilon$ in a space of analytic functions which have\na ho…
On the Cauchy problem for the Muskat equation with non-Lipschitz initial data Open
This article is devoted to the study of the Cauchy problem for the Muskat equation. We consider initial data belonging to the critical Sobolev space of functions with three-half derivative in $L^2$, up to a fractional logarithmic correctio…
Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem Open
We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first well-posed…
Endpoint Sobolev theory for the Muskat equation Open
This paper is devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. We prove that the Cauchy problem is well-posed on the endpoint Sobolev space of $L^2$ functions with three-half derivative in…
On the Cauchy problem for the Muskat equation with non-Lipschitz initial\n data Open
This article is devoted to the study of the Cauchy problem for the Muskat\nequation. We consider initial data belonging to the critical Sobolev space of\nfunctions with three-half derivative in $L^2$, up to a fractional logarithmic\ncorrec…
Cauchy theory for the water waves system in an analytic framework Open
In this paper we consider the Cauchy problem for gravity water waves, in a domain with a flat bottom and in arbitrary space dimension. We prove that if the data are of size $\varepsilon$ in a space of analytic functions which have a holomo…
Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics Open
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 lubrication approximations, the Boussinesq and…