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Strichartz inequalities with white noise potential on compact surfaces

Antoine Mouzard, Immanuel Zachhuber

Abstract

We prove Strichatz inequalities for the Schr{ö}dinger equation and the wave equation with multiplicative noise on a two-dimensional manifold. This relies on the Anderson Hamiltonian H described using high order paracontrolled calculus. As an application, it gives a low regularity solution theory for the associated nonlinear equations.

Strichartz inequalities with white noise potential on compact surfaces

Abstract

We prove Strichatz inequalities for the Schr{ö}dinger equation and the wave equation with multiplicative noise on a two-dimensional manifold. This relies on the Anderson Hamiltonian H described using high order paracontrolled calculus. As an application, it gives a low regularity solution theory for the associated nonlinear equations.

Paper Structure

This paper contains 11 sections, 245 equations.

Table of Contents

  1. Preliminaries
  2. Strichartz inequalities on manifolds
  3. Basics on paracontrolled calculus
  4. Construction of the Anderson Hamiltonian
  5. Strichartz inequalities for the stochastic Schrödinger equation
  6. Strichartz inequalities for the Schrödinger group
  7. Local well-posedness for multiplicative stochastic cubic NLS
  8. Strichartz inequalities for the stochastic wave equation
  9. Strichartz inequalities for the wave propagator
  10. Strichartz inequalities for wave equations with rough potentials
  11. Local well-posedness for the multiplicative cubic stochastic wave equation

Theorems & Definitions (1)

  • remark 1