Minimal non-scattering solutions for the Zakharov system

Candy, Timothy

doi:10.48550/arxiv.2205.08867

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Introduction and Main Results1

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2023

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“…The scattering for general non-radial data below the ground state remains a challenging open problem. We refer to the recent progress by Candy [14] on the existence of a minimal energy almost periodic solution if the scattering fails.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The three dimensional stochastic Zakharov system

Herr¹,

Röckner²,

Spitz³

et al. 2023

Preprint

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We study the three dimensional stochastic Zakharov system in the energy space, where the Schrödinger equation is driven by linear multiplicative noise and the wave equation is driven by additive noise. We prove the well-posedness of the system up to the maximal existence time and provide a blowup alternative. We further show that the solution exists at least as long as it remains below the ground state. Two main ingredients of our proof are refined rescaling transformations and the normal form method. Moreover, in contrast to the deterministic setting, our functional framework also incorporates the local smoothing estimate for the Schrödinger equation in order to control lower order perturbations arising from the noise. Finally, we prove a regularization by noise result which states that finite time blowup before any given time can be prevented with high probability by adding sufficiently large non-conservative noise. The key point of its proof is a Strichartz estimate for the Schrödinger equation with a potential solving the free wave equation.

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Section: Introduction and Main Resultsmentioning

confidence: 99%