Wellposedness of the cubic Gross-Pitaevskii equation with spatial white noise on $\mathbb{R}^2$

Pierre Mackowiak

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arXiv · 2311.10405

Wellposedness of the cubic Gross-Pitaevskii equation with spatial white noise on $\mathbb{R}^2$

Pierre Mackowiak

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Abstract

In this paper, we prove the global wellposedness of the Gross-Pitaevskii equation with white noise potential, i.e. a cubic nonlinear Schr{\"o}dinger equation with harmonic confining potential and spatial white noise multiplicative term. This problem is ill-defined and a Wick renormalization is needed in order to give a meaning to solutions. In order to do this, we introduce a change of variables which transforms the original equation into one with less irregular terms. We construct a solution as a limit of solutions of the same equation but with a regularized noise. This convergence is shown by interpolating between a diverging bound in a high regularity Hermite-Sobolev space and a Cauchy estimate in $\mathbb{L}^2(\mathbb{R}^2)$.

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Pierre Mackowiak. 2023-11-17. Wellposedness of the cubic Gross-Pitaevskii equation with spatial white noise on $\mathbb{R}^2$. https://arxiv.org/abs/2311.10405

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