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Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Van Duong Dinh

Abstract

We consider the cubic nonlinear fourth-order Schrödinger equation \[ i\partial_t u - Δ^2 u + μΔu = \pm |u|^2 u, \quad μ\geq 0 \] on $\mathbb{R}^N, N \geq 5$ with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Abstract

We consider the cubic nonlinear fourth-order Schrödinger equation on with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

Paper Structure

This paper contains 10 sections, 17 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Known results
  4. Main results
  5. Preliminaries
  6. Notations
  7. Wiener randomization and Probabilistic Strichartz estimates
  8. Function spaces and their properties
  9. Probabilistic nonlinear estimates
  10. Probabilistic well-posedness

Key Result

Theorem \oldthetheorem

Let $N\geq 5$, $\mu \geq 0$ and $\gamma \in (\gamma_N, \mathop{\mathrm{\gamma_c}}\limits)$. Let $f\in H^\gamma(\mathbb R^N)$ and $f^\omega$ be the Wiener randomization defined in defi-random satisfying cond-distri. Then the equation 4NLS is almost surely locally well-posed with respect to the random

Theorems & Definitions (30)

  • Theorem \oldthetheorem: Almost sure local well-posedness
  • Theorem \oldthetheorem: Probabilistic small data global well-posedness and scattering
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Large probability global well-posedness and scattering
  • Lemma \oldthetheorem: BOP
  • Remark \oldthetheorem
  • Lemma \oldthetheorem: BOP
  • Remark \oldthetheorem
  • Lemma \oldthetheorem: BaKS
  • Lemma \oldthetheorem: Strichartz estimates Pausader-DPDE
  • ...and 20 more