Related papers: Invariant Gibbs measure evolution for the radial n…
We establish new results for the radial nonlinear wave and Schr\"odinger equations on the ball in $\Bbb R^2$ and $\Bbb R^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the…
We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schr\"odinger (NLS) and nonlinear wave (NLW) equations on the unit ball in R^d to the case…
In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In…
Our first purpose is to extend the results from \cite{T} on the radial defocusing NLS on the disc in $\mathbb{R}^2$ to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs…
In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The…
We study the long time behavior of the subcritical (subcubic) defocussing nonlinear wave equation on the three dimensional ball, for random data of low regularity. We prove that for a large set of radial initial data in $\cap_{s<1/2}…
We consider the defocusing nonlinear Schr\"odinger equation on $\mathbb{T}^2$ with Wick ordered power nonlinearity, and prove almost sure global well-posedness with respect to the associated Gibbs measure. The heart of the matter is the…
Nonlinear waves in a liquid containing gas bubbles are considered in the three-dimensional case. Nonlinear evolution equation is given for description of long nonlinear pressure waves. It is shown that in the general case the equation is…
The dynamics of gravitational waves is investigated in full 3+1 dimensional numerical relativity, emphasizing the difficulties that one might encounter in numerical evolutions, particularly those arising from non-linearities and gauge…
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schroedinger equations on the disc of the plane $\R^2$. We also prove an estimate giving some intuition to what may happen in 3…
We consider the two-dimensional defocusing nonlinear Schr\"odinger equation (NLS) on the unit disc in the plane with the Gibbs initial data under radial symmetry. By using a type of random averaging operator ansatz, we build a strong…
We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic $\Phi^4_3$-model. This result is the hyperbolic counterpart to seminal works on the…
We study the evolution equations for gravitational waves, which are derived using the full metric to raise and lower indices. This method ensures full consistency between the Ricci tensor and all gauge restrictions and requirements, and…
We consider a fractional nonlinear wave equations (fNLW) with a general power-type nonlinearity, on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for a renormalized fNLW. We first construct…
We consider the one dimensional cubic nonlinear Schr{\"o}dinger equation with trapping potential behaving like |x| s (s > 1) at infinity. We construct Gibbs measures associated to the equation and prove that the Cauchy problem is globally…
In this paper, we build a Gibbs measure for the cubic defocusing Schr\"odinger equation on the real line with a decreasing interaction potential, in the sense that the non linearity $|u|^2u$ is multiplied by a function $\chi$ which we…
In this note, we give an overview of some results obtained in [3]. This latter work is devoted to the study of the one-dimensional nonlinear Schr{\"o}dinger equation with random initial conditions. Namely, we describe the nonlinear…
We show, by the means of several examples, how we can use Gibbs measures to construct global solutions to dispersive equations at low regularity. The construction relies on the Prokhorov compactness theorem combined with the Skorokhod…
We consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick…
In this article, we first present the construction of Gibbs measures associated to nonlinear Schr\"odinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial…