67 research outputs found
On the cubic NLS on 3D compact domains
We prove bilinear estimates for the Schr\"odinger equation on 3D domains,
with Dirichlet boundary conditions. On non-trapping domains, they match the
case, while on bounded domains they match the generic boundary
less manifold case. As an application, we obtain global well-posedness for the
defocusing cubic NLS for data in , , with
any bounded domain with smooth boundary.Comment: 15 pages, updated references and corrected typos. To appear in
Journal of the Institute of Mathematics of Jussie
On uniqueness for the critical wave equation
We prove the uniqueness of weak solutions to the critical defocusing wave
equation in 3D under a local energy inequality condition. More precisely, we
prove the uniqueness of , under the condition that verifies some local
energy inequalities.Comment: 12 pages, to appear in Comm. Partial Differential Equation
Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case
We consider a model case for a strictly convex domain of dimension
with smooth boundary and we describe dispersion for the wave equation with
Dirichlet boundary conditions. More specifically, we obtain the optimal fixed
time decay rate for the smoothed out Green function: a loss occurs
with respect to the boundary less case, due to repeated occurrences of
swallowtail type singularities in the wave front set.Comment: 53 pages, 4 figures, to appear in Annals of Math. Fixed typos, added
remark
A profile decomposition approach to the L∞/t (L3/ x) Navier–Stokes regularity criterion
In this paper we continue to develop an alternative viewpoint on recent studies of Navier–Stokes regularity in critical spaces, a program which was started in the recent work by Kenig and Koch (Ann Inst H Poincaré Anal Non Linéaire 28(2):159–187, 2011). Specifically, we prove that strong solutions which remain bounded in the space L3(R3) do not become singular in finite time, a known result established by Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003) in the context of suitable weak solutions. Here, we use the method of “critical elements” which was recently developed by Kenig and Merle to treat critical dispersive equations. Our main tool is a “profile decomposition” for the Navier–Stokes equations in critical Besov spaces which we develop here. As a byproduct of this tool, assuming a singularity-producing initial datum for Navier–Stokes exists in a critical Lebesgue or Besov space, we show there is one with minimal norm, generalizing a result of Rusin and Sverak (J Funct Anal 260(3):879–891, 2011)
Square function and heat flow estimates on domains
The first purpose of this note is to provide a proof of the usual square
function estimate on Lp (?). It turns out to follow directly from a generic
Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on
Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker
version of the square function estimate, which is enough in most instances
involving dispersive PDEs. Moreover, we obtain, by a relatively simple
integration by parts, several useful Lp (?; H) bounds for the derivatives of
the heat ?ow with values in a given Hilbert space H
Bilinear virial identities and applications
We prove bilinear virial identities for the nonlinear Schrodinger equation,
which are extensions of the Morawetz interaction inequalities. We recover and
extend known bilinear improvements to Strichartz inequalities and provide
applications to various nonlinear problems, most notably on domains with
boundaries.Comment: 30 pages, final version to appear in the Annales Scientifiques de
l'EN
Self-improving bounds for the Navier-Stokes equations
We consider regular solutions to the Navier-Stokes equation and provide an
extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative
regularity Besov scale, with regularity arbitrarly close to -1. Our results
rely on turning a priori bounds for the solution in negative Besov spaces into
bounds in the positive regularity scale.Comment: 11 pages, updated references, to appear in Bull. Soc. Math. Franc
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