Searcharxiv⌕ Search

arXiv subjects

Claude Zuily

Publications and source records attributed to Claude Zuily.

At least 19 recordsLinked to original sources
arXiv2024

Nonlinear interpolation and the flow map for quasilinear equations

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.

math.AP↗
arXiv2023

Domains of dependence for subelliptic wave equations and unique continuation for fractional powers of Hörmander's operators

We prove the sharp domain of dependence property for solutions to subelliptic wave equations for sums of squares of vector fields satisfying Hörmander bracket condition. We deduce a unique continuation property for the square root of subelliptic Laplace operators under an additional analyticity condition. Then, with a different, more involved method, we prove the same result of unique continuation for more general $s$-powers ($0<s<1$).

math.AP↗
arXiv2023

Virial theorems and equipartition of energy for water-waves

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.

math.AP↗
arXiv2021

A remark on quantitative unique continuation from subsets of the boundary of positive measure

The question of unique continuation of harmonic functions in a domain $Ω$ $\subset$ R d with boundary $\partial$$Ω$, satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset $ω$ of the boundary is a classical problem. When $ω$ contains an open subset of the boundary it is a consequence of Carleman estimates (uniqueness for second order elliptic operators across an hypersurface). The case where $ω$ is a set of positive (d -- 1) dimensional measure has attracted a lot of attention, see e.g. [10, 3, 15], where qualitative results have been obtained in various situations. The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [13, 14], we consider the simpler case of W 2,$\infty$ domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2,$\infty$ domain with boundary.

math.AP↗
arXiv2021

Cauchy theory for the water waves system in an analytic framework

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 $σ$, 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 $σ- C'\varepsilon t$.

math.AP↗
arXiv2016

Real analyticity of radiation patterns of resonant states on asymptotically hyperbolic manifolds

We show that resonant states in scattering on asymptotically hyperbolic man-ifolds that are analytic near conformal infinity, have analytic radiation patterns at infinity. On even asymptotically hyperbolic manifolds we also show that smooth solutions of Vasy operators with analytic coefficients are also analytic. That answer a question of M.Zworski ([14] Conjecture 2). The proof is based on previous results of Baouendi-Goulaouic and Bolley-Camus-Hanouzet and for convenience of the reader we present an outline of the proof of the latter.

math.AP↗
arXiv2015

Laplace Eigenfunctions And Damped Wave Equation Ii: Product Manifolds

- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies (modulo rescalling) and deduce new stabilization results for weakly damped wave equations which extend to product manifolds previous results by Leautaud-Lerner [12] obtained for products of tori.

math.AP↗
arXiv2014

On the Cauchy problem for gravity water waves

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of~$C^{3/2+ε}$-class for some $ε>0$ and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.

math.AP↗
arXiv2014

Cauchy theory for the gravity water waves system with non localized initial data

In this article, we develop the local Cauchy theory for the gravity water waves system, for rough initial data which do not decay at infinity. We work in the context of $L^2$-based uniformly local Sobolev spaces introduced by Kato. We prove a classical well-posedness result (without loss of derivatives). Our result implies also a local well-posedness result in Hölder spaces (with loss of $d/2$ derivatives). As an illustration, we solve a question raised by Boussinesq on the water waves problem in a canal. We take benefit of an elementary observation to show that the strategy suggested by Boussinesq does indeed apply to this setting.

math.AP↗
arXiv2014

Strichartz estimates and the Cauchy problem for the gravity water waves equations

This paper is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, we can consider solutions such that the curvature of the initial free surface does not belong to $L^2$. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and Hölder estimates. We first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then we use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.

math.AP↗
arXiv2013

Strichartz estimates for gravity water waves

We prove Strichartz estimates for gravity water waves, in arbitrary dimension and in fluid domains with general bottoms. We consider rough solutions such that, initially, the first order derivatives of the velocity field are not controlled in $L^\infty$-norm, or the initial free surface has a curvature not controlled in $L^2$-norm.

math.AP↗
arXiv2012

The water waves equations: from Zakharov to Euler

Starting form the Zakharov/Craig-Sulem formulation of the water-waves equations, we prove that one can define a pressure term and hence obtain a solution of the classical Euler equations. It is proved that these results hold in rough domains, under minimal assumptions on the regularity to ensure, in terms of Sobolev spaces, that the solutions are $C^1$.

math.AP↗
arXiv2010

Strichartz Estimates for Water Waves

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [2]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at (? = 0, ? = 0)).

math.AP↗
arXiv2009

On the Water Waves Equations with Surface Tension

The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developped by T. Alazard and G. Métivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schrödinger type. We then show that the smoothing effect for the (one dimensional) surface tension water waves proved by H. Christianson, V. M. Hur, and G. Staffilani in [9], is in fact a rather direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.

math.AP↗