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arXiv · 2410.05034

The energy-critical stochastic Zakharov system

Sebastian Herr·Michael Röckner·Martin Spitz·Deng Zhang
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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 $H^1\times L^2$ 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.

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Sebastian Herr, Michael Röckner, Martin Spitz, Deng Zhang. 2024-10-07. The energy-critical stochastic Zakharov system. https://arxiv.org/abs/2410.05034

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