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Global well-posedness of the 4-d energy-critical stochastic nonlinear Schrödinger equations with non-vanishing boundary condition

Kelvin Cheung, Guopeng Li

Abstract

We consider the energy-critical stochastic cubic nonlinear Schrödinger equation on $\mathbb R^4$ with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrödinger equation on $\mathbb R^4$, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.

Global well-posedness of the 4-d energy-critical stochastic nonlinear Schrödinger equations with non-vanishing boundary condition

Abstract

We consider the energy-critical stochastic cubic nonlinear Schrödinger equation on with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrödinger equation on , we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.

Paper Structure

This paper contains 14 sections, 8 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Stochastic nonlinear schrödinger equation
  3. Main result
  4. Preliminaries
  5. Strichartz estimates
  6. On the stochastic convolution
  7. Perturbation lemma
  8. Energy-critical NLS with a perturbation
  9. Local well-posedness of the perturbed NLS
  10. Global existence of solutions to the perturbed NLS
  11. Proof of the main theorem
  12. Bound on the energy
  13. Global existence of SNLS
  14. Unconditional uniqueness

Key Result

Theorem 1.1

Let $\phi \in \textup{HS}(L^2(\mathbb{R}^4); H^1(\mathbb{R}^4))$. Then, the SNLS SNLS with the condition Non-VB is globally well-posed in the energy space $\mathcal{E}(\mathbb{R}^4)$. In particular, solutions are unique in the class $\Psi+ C( \mathbb{R}_+;\mathcal{E}(\mathbb{R}^4))$.

Theorems & Definitions (14)

  • Theorem 1.1: Global well-poseness of SNLS
  • Remark 1.2
  • Lemma 2.1: Strichartz estimates
  • Lemma 2.2
  • Lemma 2.3: Perturbation lemma
  • Remark 2.4
  • Proposition 3.1: Local well-posedness of the perturbed NLS
  • Remark 3.2
  • proof
  • Proposition 3.3
  • ...and 4 more