Claude Zuily

On the water-wave 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. Metivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schrodinger 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.

Microlocal analytic smoothing effect for the Schrödinger equation

0. Introduction and main results. The purpose of this work is to study the microlocal analytic smoothness of solutions of the initial value problem for the linear Schrödinger equation with variable coefficients. The aim is to relate the behavior at infinity of the initial data with the microlocal analytic smoothness; this phenomenon is known as themicrolocal smoothing effect. The results presented here are extensions to the analytic case of those of Craig-Kappeler-Strauss [CKS], [C], which concern theC∞ case. However, our method of proof, which relies on Sjöstrand theory [Sj], is entirely different from that of [CKS]. This question has also been investigated in recent years in the papers of Shananin [Sh], Kapitanski-Safarov [KS], and Wunsch [W]. Related results have also been obtained by Doi [D1], [D2], and we refer to the paper [CKS] for more references on the subject. Let us describe our main result. Let P = P(y,Dy), a second-order differential operator inR,

Remark on the Kato Smoothing Effect for Schrödinger Equation with Superquadratic Potentials

The aim of this note is to extend recent results of Yajima and Zhang (2001, 2004) on the -smoothing effect for Schrödinger equation with potential growing at infinity faster than quadratically.

Unicite et non unicite du probleme de cauchy pour des operateurs hyperboliques a characteristiques doubles

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Existence locale de solutions c∞ pour des equations de monge-ampere changeant de type

We prove the C∞ local solvability of the n-dimensional Monge-A mpe:ere equation det (uij + aij(x,u,▽u))= K(x) f(x, u, ▽u), f> 0, in a neighborhood of any point x0 where K(x0)= 0 but dK(xo) ¦0.

Uniqueness and Pseudo-Convexity

In the results described before, no condition (except to be non-characteristic) was imposed on the initial hypersurface and in particular, uniqueness did not depend on the side containing the support of the Solution; however, precise hypotheses like smoothness, muitiplicity of the characteristic roots, were made. In the general case (where no smoothness occurs) we shall see that uniqueness depends on geometrical conditions between the operator and the hypersurface, called “pseudo-convexity conditions”.

Calderon’s Theorem and Its Extensions

In a neighborhood V of a point xo in ℝn let be a C∞ hypersurface and a differential operator of order m whose principal symbol pm(x, ξ) has C∞ coefficients in V.