Carlos E. Kenig

A bilinear estimate with applications to the KdV equation

u(x, 0) = u0(x), where u0 ∈ H(R). Our principal aim here is to lower the best index s for which one has local well posedness in H(R), i.e. existence, uniqueness, persistence and continuous dependence on the data, for a finite time interval, whose size depends on ‖u0‖Hs . Equation in (1.1) was derived by Korteweg and de Vries [21] as a model for long wave propagating in a channel. A large amount of work has been devoted to the existence problem for the IVP (1.1). For instance, (see [9], [10]), the inverse scattering method applies to this problem, and, under appropriate decay assumptions on the data, several existence results have been established, see [5],[6],[14],[28],[33]. Another approach, inherited from hyperbolic problems, relies on the energy estimates, and, in particular shows that (1.1) is locally well posed in H(R) for s > 3/2, (see [2],[3],[12],[29],[30],[31]). Using these results and conservation laws, global (in time) well posedness in H(R), s ≥ 2 was established, (see [3],[12],[30]). Also, global in time weak solutions in the energy space H(R) were constructed in [34]. In [13] and [22] a “local smoothing” effect for solutions of (1.1) was discovered. This, combined with the conservation laws, was used in [13] and [22] to construct global in time weak solutions with data in H(R), and even in L(R). In [16], we introduced oscillatory integral techniques, to establish local well posedness of (1.1) in H(R), s > 3/4, and hence, global (in time) well posedness in H(R), s ≥ 1. (In [16] we showed how to obtain the above mentioned result by Picard iteration in an appropriate function space.) In [4] J. Bourgain introduced new function spaces, adapted to the linear operator ∂t+∂ 3 x, for which there are good “bilinear” estimates for the nonlinear term ∂x(u /2). Using these spaces, Bourgain was able to establish local well posedness of (1.1) in H(R) = L(R), and hence, by a conservation

Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems

Introduction Divergence form elliptic equations Some classes of examples and their perturbation theory Epilogue: Some further results and open problems References.

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.

Well-posedness of the initial value problem for the Korteweg-de Vries equation

(1.1) &ItU + axu + U1xU = O, x, t E R { u(x, 0) = uo(x). The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been considered in different contexts, namely in its relation with the inverse scattering method, in plasma physics, and in algebraic geometry (see [24], and references therein). Our purpose is to study local and global well-posedness of the IVP (1.1) in classical Sobolev spaces Hs(R) . We shall say that the IVP (1.1) is locally (resp. globally) well-posed in the function space X if it induces a dynamical system on X by generating a continuous local (resp. global) flow. It was established in the works of Bona and Smith [3], Bona and Scott [2], Saut and Temam [30], and Kato [ 1 5] that the IVP (1. 1) is locally (resp. globally) well-posed in Hs with s > 3/2 (resp. s > 2). Roughly speaking, global well-posedness in Hs depends on the available local theory and on the conservation laws satisfied by solutions of (1.1), namely:

The Neumann problem on Lipschitz domains

Au — 0 in D; u = ƒ on bD9 where ƒ and its gradient on 3D belong to L(do). For C domains, these estimates were obtained by A. P. Calderón et al. [1]. For dimension 2, see (d) below. In [4] and [5] we found an elementary integral formula (7) and used it to prove a theorem of Dahlberg (Theorem 1) on Lipschitz domains. Unknown to us, this formula had already been discovered long ago by Payne and Weinberger and applied to the Dirichlet problem in smooth domains. Moreover, they used a second formula (2), which is a variant of a formula due to F. Rellich [7], to study the Neumann problem in smooth domains. We show here that the same strategy as in [4] applied to the second formula (2) coupled with Dahlbergs theorem yields our main result. Thus integral formulas give appropriate estimates for the solution of not only the Dirichlet problem, but also the Neumann problem on Lipschitz domains. We will present a more general version that applies to variable coefficient operators, systems, and other elliptic problems in a later

Hardy spaces and the Neumann problem in L^p for laplace's equation in Lipschitz domains

On donne des resultats optimaux pour la resolubilite du probleme de Neumann dans des domaines de Lipschitz a donnees dans L p . On obtient des resultats du point extreme correspondant pour des espaces de Hardy

Global well-posedness of the Benjamin–Ono equation in low-regularity spaces

We prove that the Benjamin-Ono initial value problem is globally well-posed in the Sobolev spaces

Small solutions to nonlinear Schrödinger equations

H^\sigma_r

Limiting Carleman weights and anisotropic inverse problems

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Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

\sigma\geq 0