arXiv subjects
Publications and source records attributed to Thomas Alazard.
The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense $G_\delta$ set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.
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 well-posed in time for any irrotational initial data in the energy space, without any smallness assumption. We also prove continuity with respect to the initial data and the propagation of regularity. The main novelty is that no dissipative effect is assumed in the system. In the absence of parabolic regularization, the key observation is that the system can be transformed into a nonlinear Schr\"odinger-type equation, to which dispersive estimates are applied. This allows us to construct solutions that are very rough from the point of view of fluid dynamics-the initial fluid velocity has merely one-half derivative in $L^2$. The main difficulty is that the problem is critical in the energy space with respect to several key inequalities from harmonic analysis. The proof incorporates new estimates for the Dirichlet-to-Neumann operator in the low-regularity regime, including refinements of paralinearization formulas and shape derivative formulas, which played a key role in the analysis of water waves.
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 invertibility of linearized operators that arise in a Nash-Moser/KAM type scheme. The goal of this paper is to prove paradifferential reducibility results, enabling the reduction of nonlinear equations themselves, rather than just their linearizations, to constant coefficient form, modulo smoothing terms. As an initial application, we demonstrate the existence of quasiperiodic solutions for certain hyperbolic systems. Despite the small denominator problem, our proof does not rely on traditional Nash-Moser/KAM-type schemes, but instead on the Banach fixed point theorem. To achieve this, we develop two key toolsets. The first focuses on the calculus of paracomposition operators introduced by Alinhac, interpreted as the flow map of a paraproduct vector field. We refine this approach to establish new estimates that precisely capture the dependence on the diffeomorphism in question. The second toolset addresses two classical reducibility problems, one for matrix differential operators and the other for nearly parallel vector fields on torus. We resolve these problems by paralinearizing the conjugacy equation and exploiting, at the paradifferential level, the specific algebraic structure of conjugacy problems, akin to Zehnder's approximate Nash-Moser approach.
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 introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results about interpolation theory. It depends solely on the concepts of Friedrichs' mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data to solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.
We consider the Cauchy problem for an inviscid irrotational fluid on a domain with a free boundary governed by a fourth order linear elasticity equation. We first derive the Craig-Sulem-Zakharov formulation of the problem and then establish the existence of a global weak solution in two space dimensions, in the general case without a damping term, for any initial data with finite energy.
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 equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. These equations are different in nature, dispersive or parabolic, but we will see that they can be studied using related tools. In particular, we will discuss a paradifferential approach to these problems.
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 in the topology of the fluid domain) and which satisfies several properties: We provide simple comparison arguments, establish a new stability estimate and derive several consequences, including monotonicity and continuity results for the solutions, along with many Lyapunov functionals. We establish an eventual analytic regularity result for any arbitrary initial data. We also study numerous qualitative properties, including global regularity for initial data in sub-critical Sobolev spaces, well-posedness in a strong sense for initial data with barely a modulus of continuity, as well as waiting-time phenomena for Lipschitz solutions, in any dimension. This revision contains some corrections.
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 greatly simplifies KAM proofs, renders the traditional KAM iteration steps unnecessary, and may suggest a systematic scheme of finding quasi-periodic solutions of realistic magnitude.
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 the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.
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 identity for the complete nonlinear water wave problem, which is valid for arbitrary solutions. As an application, we obtain non-perturbative results justifying the formation of bubbles for the free-surface Rayleigh-Taylor instability, for any non-zero initial data. We also derive exact virial identities involving higher order energies. The fact that such exact identities are valid for nonlinear equations is new and general: as explained in a companion paper, similar identities can be derived for many other nonlinear equations. We illustrate this result by an explicit computation for standing waves. As side results, we prove trace inequalities for harmonic functions in Lipschitz domains which are optimal with respect to the dependence in the Lipschitz norm of the graph.
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 Lipschitz. For two-dimensional domains, we obtain a sharp $L^p$-estimate for $1<p\leq 2$ by using a Riemann mapping and interpolation argument.
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 model equations include a convection diffusion equation for the polarity marker concentration and the incompressible Darcy's equation. The mathematical novelty of this problem is the nonlinear destabilizing term in the boundary condition that describes the active character of the cell cytoskeleton. We first study the linear stability of this problem and we show that, above a well precise threshold, the disk becomes linearly unstable. By using two different approaches we prove existence of traveling wave solutions, which describes persistent motion of a biological cell. One is explicit, by construction. The other is established implicitly, as the one bifurcating from stationary solution.
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-Belinch\'on 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 $H^s(\mathbb{S})$ with $s>3/2$. In this paper, we consider the Cauchy problem in the critical space $H^{1/2}(\mathbb{S})$. Two interesting new features, at this level of regularity, are that the equation can be written in the form $$ \partial_t u+V\partial_x u+\gamma \Lambda u=0, $$ where $\gamma$ is non-negative but not bounded from below and $V/\sqrt{\gamma}$ 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 $H^{1/2}(\mathbb{S})$. 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.
We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in $L^2$. Moreover, we prove that the solution exists globally in time under a smallness assumption.
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-posedness result for arbitrary large data in the critical space $\dot{H}^2(\mathbb{R}^2)\cap W^{1,\infty}(\mathbb{R}^2)$. Moreover, we prove the existence of solutions for initial data which are not Lipschitz.
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 $L^2$. This result is optimal with respect to the scaling of the equation. One well-known difficulty is that one cannot define a flow map such that the lifespan is bounded from below on bounded subsets of this critical Sobolev space. To overcome this, we estimate the solutions for a norm which depends on the initial data themselves, using the weighted fractional Laplacians introduced in our previous works. Our proof is the first in which a null-type structure is identified for the Muskat equation, allowing to compensate for the degeneracy of the parabolic behavior for large slopes.
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 correction. As a corollary, we obtain the first local and global well-posedness results for initial free surface which are not Lipschitz.
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 holomorphic extension in a strip of size $\sigma$, then the solution exists up to a time of size $C/\varepsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma - C'\varepsilon t$.