On Uniqueness of Multi-bubble Blow-Up Solutions and Multi-solitons to $$L^2$$-Critical Nonlinear Schrödinger Equations

Cao, Daomin; Su, Yiming; Deng, Zongquan

doi:10.1007/s00205-022-01832-x

Cited by 5 publications

(1 citation statement)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also refer to [22] for the construction of stochastic log-log blow-up solutions, and to [46,47,54] for the construction and conditional uniqueness of multi-bubble Bourgain-Wang type blow-up solutions and (non-pure) multi-solitons, which provide new examples for the mass quantization conjecture and the soliton resolution conjecture. For the refined uniqueness of pure multi-solitons in the low asymptotical regime, related to an open problem raised by Martel [40], we refer to [13].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The three dimensional stochastic Zakharov system

Herr¹,

Röckner²,

Spitz³

et al. 2023

Preprint

View full text Add to dashboard Cite

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.

show abstract