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Local and global well-posedness of 2d periodic multiplicative stochastic NLS

Immanuel Zachhuber

Abstract

We use Strichartz estimates with rough potentials like the spatial white noise on the 2 \ dimensional torus to prove global well-posedness of the multiplicative stochastic NLS with general integer powers in both the energy and strong regime together with controls over the growth of the norms of the solutions.

Local and global well-posedness of 2d periodic multiplicative stochastic NLS

Abstract

We use Strichartz estimates with rough potentials like the spatial white noise on the 2 \ dimensional torus to prove global well-posedness of the multiplicative stochastic NLS with general integer powers in both the energy and strong regime together with controls over the growth of the norms of the solutions.

Paper Structure

This paper contains 10 sections, 32 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Classical Strichartz estimates on the torus
  3. The Anderson Hamiltonian in 2 dimensions
  4. Strichartz estimates for the Anderson Hamiltonian
  5. Well-posedness of multiplicative stochastic NLS
  6. Low-regularity solutions
  7. Global well-posedness in the energy space
  8. Global well-posedness of strong solutions and growth of norms
  9. Paracontrolled Distributions and Besov spaces
  10. Almost adjointness lemmas for paraproducts

Key Result

Theorem 1.1

[2d Anderson Strichartz Estimates]Let $r \geqslant 4$, then we have for any $\delta > 0$

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 30 more