Random data for Schrödinger equations : construction of global solutions for supercritical equations
type: Dataset
description: In this thesis, we build a large number of global solutions for many supercritical Schrödinger equations. The method is to make the random initial data, using the same methods that Nicolas Burq, Nikolay Tzvetkov and Laurent Thomann in order to obtain differentiability. First, we consider the cubic Schrödinger equation in three dimensional. Using Gaussian random variables and the basis of L^2(R^3) consists of tensorial Hermite functions, we construct sets of solutions for initial data that are morally in L^2(R^3). The main ingredients of the proof are the existence of Bourgain type bilinear estimates for the harmonic oscillator and the lens transform which can be reduced to prove a local existence of solutions for the Schrödinger equation with harmonic potential. Next, we study the smoothing effect to prove an analogous theorem which the gain of differentiability is equalto 1/2-2/(p-1) which p is the nonlinearity of the equation. This gain is lower than previously but the basis of eigenfunctions are general. As the method uses only linear estimates, we establish the result for a general class of random variables.Finally, we prove multilinear estimates in two dimensional for a basis of ordinaries eigenfunctions and Wienerchaos type inequalities for classical random variables. This allows us to establish the theorem for the quinticSchrödinger equation, with a gain of differentiability equals to 1/3, in the same context as the previous chapter.
landing page: https://theses.hal.science/tel-00771354
identifier: NNT: 2012PA112333
keyword: bourgain-type-bilinear-estimates
keyword: donnees-aleatoires
keyword: equations-de-schrodinger-sur-critiques
keyword: estimees-bilineaires-de-type-bourgain
keyword: global-solutions
keyword: harmonic-oscillator
keyword: infoeu-reposemanticsdoctoralthesis
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: oscillateur-harmonique
keyword: random-data
keyword: solutions-globales
keyword: supercritical-non-linear-schrodinger-equations
keyword: theses
issued: 2026-05-15T12:15:53.186653
modified: 2026-05-15T12:15:53.186668
language: fr
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

Random data for Schrödinger equations : construction of global solutions for supercritical equations
type: Distribution
format: HTML
access URL: https://theses.hal.science/tel-00771354
issued: 2026-05-15T12:15:53.240938
modified: 2026-05-15T12:15:53.145449