Random data Cauchy theory for supercritical wave equations I: local theory

N Burq, N Tzvetkov - Inventiones mathematicae, 2008 - Springer
We study the local existence of strong solutions for the cubic nonlinear wave equation with
data in H s (M), s< 1/2, where M is a three dimensional compact Riemannian manifold. This …

Ill-posedness of the Navier–Stokes equations in a critical space in 3D

J Bourgain, N Pavlović - Journal of Functional Analysis, 2008 - Elsevier
We prove that the Cauchy problem for the three-dimensional Navier–Stokes equations is ill-
posed in B˙∞− 1,∞ in the sense that a “norm inflation” happens in finite time. More …

On fractional Schrodinger equations in Sobolev spaces

Y Hong, Y Sire - arXiv preprint arXiv:1501.01414, 2015 - arxiv.org
Let $\sigma\in (0, 1) $ with $\sigma\neq\frac {1}{2} $. We investigate the fractional nonlinear
Schr\" odinger equation in $\mathbb R^ d $: $$ i\partial_tu+ (-\Delta)^\sigma u+\mu| u|^{p-1} …

Probabilistic well-posedness for the cubic wave equation

N Burq, N Tzvetkov - Journal of the European Mathematical Society, 2013 - ems.press
The purpose of this article is to introduce for dispersive partial differential equations with
random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We …

The cubic fourth-order Schrödinger equation

B Pausader - Journal of Functional Analysis, 2009 - Elsevier
Fourth-order Schrödinger equations have been introduced by Karpman and Shagalov to
take into account the role of small fourth-order dispersion terms in the propagation of intense …

Random Data Cauchy Theory for Nonlinear Wave Equations of Power-Type on ℝ3

J Lührmann, D Mendelson - Communications in Partial Differential …, 2014 - Taylor & Francis
We consider the defocusing nonlinear wave equation of power-type on ℝ3. We establish an
almost sure global existence result with respect to a suitable randomization of the initial …

Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions

T Tao, M Visan, X Zhang - 2007 - projecteuclid.org
We establish global well-posedness and scattering for solutions to the defocusing mass-
critical (pseudoconformal) nonlinear Schrödinger equation iu t+ Δ u=| u| 4/nu for large …

A symmetric low-regularity integrator for nonlinear Klein-Gordon equation

Y Wang, X Zhao - Mathematics of Computation, 2022 - ams.org
In this work, we propose a symmetric exponential-type low-regularity integrator for solving
the nonlinear Klein-Gordon equation under rough data. The scheme is explicit in the …

Sharp well-posedness for the cubic NLS and mKdV in

B Harrop-Griffiths, R Killip, M Vişan - Forum of Mathematics, Pi, 2024 - cambridge.org
We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is
globally well-posed in (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of …

Invariant measures for the defocusing nonlinear Schrödinger equation

N Tzvetkov - Annales de l'Institut Fourier, 2008 - numdam.org
In [12], we constructed and proved the invariance of a Gibbs measure associated to the sub-
cubic, focusing or defocusing Nonlinear Schrödinger equation (NLS) on the disc of the plane …