The energy-critical stochastic Zakharov system

2024/10/07 by Herr, Sebastian, Röckner, Michael, Spitz, Martin +1 · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.05034

Abstract

This work is devoted to the stochastic Zakharov system in dimension four, which is the energy-critical dimension. First, we prove local well-posedness in the energy space H¹× L² up to the maximal existence time and a blow-up alternative. Second, we prove that for large data solutions exist globally as long as energy and wave mass are below the ground state threshold. Third, we prove a regularization by noise phenomenon: the probability of global existence and scattering goes to one if the strength of the (non-conservative) noise goes to infinity. The proof is based on the refined rescaling approach and a new functional framework, where both Fourier restriction and local smoothing norms are used as well as a (uniform) double endpoint Strichartz and local smoothing inequality for the Schrödinger equation with certain rough and time dependent lower order perturbations.

Cited by

• Multi-solitons to focusing mass-supercritical stochastic nonlinear Schrödinger equations
• Regularization by noise for the energy- and mass-critical nonlinear Schrödinger equations

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