Strichartz inequalities with white noise potential on compact surfaces

Abstract

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

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

“…The authors in [ 18 , 31 , 43 ] introduced another approach to the study of the NLS ( 1.1 ) with . Their method is based on the realization of the (formal) Anderson Hamiltonian as a self-adjoint operator on the space.…”

“…Their method is based on the realization of the (formal) Anderson Hamiltonian as a self-adjoint operator on the space. Specifically, [ 18 ] considered the equation on the torus, [ 31 ] considered a compact manifold, and [ 43 ] considered the full space. In their settings, the initial data needs to belong to the domain of H .…”