Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

Gavin Stewart

Abstract

We consider a periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by \begin{equation*} idu = Δu \circ dW_t + |u|^{p-1}u\;dt \end{equation*} By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with $L^2$ initial data for nonlinearities with exponent $1 < p \leq 3$. By generalizing the Fourier restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation ($p=5$), and show that it is analytically illposed in $L^1_ωC_t L^2_x$.

On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

Abstract

We consider a periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by \begin{equation*} idu = Δu \circ dW_t + |u|^{p-1}u\;dt \end{equation*} By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with initial data for nonlinearities with exponent . By generalizing the Fourier restriction spaces to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation (), and show that it is analytically illposed in .
Paper Structure (7 sections, 11 theorems, 83 equations)

This paper contains 7 sections, 11 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Plan of the proof and obstacles
  4. Notation and Preliminaries
  5. Strichartz estimates
  6. Global wellposedness for p <= 3
  7. L2 illposedness for p = 5

Key Result

Theorem 1

Let $u_0 \in L^4_\omega L^2_x$, $1 < p \leq 3$. Then, eqn:WNDNLS admits a unique global solution $u \in L^4_\omega L^4_{t,\textup{loc}}L^4_x$. Moreover, this solution $u$ is almost surely continuous in $L^2$ with conserved $L^2$ norm.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Remark 1
  • ...and 9 more