Periodic nonlinear Schrödinger equation and invariant measures

Abstract

In this paper we continue some investigations on the periodic NLSE ίu t + u xx + u\u\ p~2 = 0 (p ^ 6) started in [LRS]. We prove that the equation is globally wellposed for a set of data φ of full normalized Gibbs measure e -βff(Φ)Hdφ(x) 9 H(φ) = \ / \φ'\ 2 -± / \φ\ p (after suitable L 2 -truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [Bl] on periodic NLS and KdV type equations.

Random data Cauchy theory for supercritical wave equations II: a global existence result

“…We decided not to pursue this issue here since our main concern in the present paper is to establish random data Cauchy theory for supercritical problems. We refer to [2,3,7,10,11,12] for results concerning the existence of invariant Gibbs measures in the closely related context of the Nonlinear Schrödinger equation.…”

“…The goal of this article is to show that in a very particular case we can combine this local theory with some invariant measure arguments (see the work by Bourgain [2,3] and the authors [10,11,5]) to obtain global solutions. Namely, we shall consider the nonlinear wave equation with Dirichlet boundary condition posed on Θ, the unit ball of R 3 , (1.1) (∂ 2 t − ∆)w + |w| α w = 0, (w, ∂ t w)| t=0 = (f 1 , f 2 ), u | Rt×∂Θ = 0, α > 0 with radial real valued initial data (f 1 , f 2 ).…”

“…where (e n (r)) ∞ n=1 is the orthonormal basis consisting in radial eigenfunctions of the Laplace operator with Dirichlet boundary conditions, associated to eigenvalues (πn) 2 .…”

Randomization and the Gross–Pitaevskii Hierarchy

“…In particular, randomization techniques have been applied in the context where the low regularity techniques of the high-low method as in the work of Bourgain [26] or the I-method, as in the work of Colliander, Keel, Staffilani, Takaoka, and Tao [60] are known not to work. As was mentioned in the introduction, this probabilistic approach was first used by Bourgain [22][23][24][25] and it has its origins in previous work of Lebowitz, Rose, and Speer [114], and Zhidkov [165]. The main idea is that global existence can be studied by means of the existence of a Gibbs measure and its invariance under the flow.…”

“…In particular, we randomize the Fourier coefficients by multiplying them by a sequence of independent identically distributed standard Bernoulli random variables (meaning that their expected value is equal to 0 and their standard deviation is equal to 1). In the nonlinear dispersive equation literature, this idea was first applied in the work of Bourgain [22][23][24][25] on almost-sure well-posedness theory for the nonlinear Schrödinger equation in low regularities. These works build on a wide range of techniques on randomization in nonlinear dispersive equations, which were first developed in the work of Lebowitz, Rose and Speer [114], and Zhidkov [165].…”