On the multi-bubble blow-up solutions to rough nonlinear Schr\"odinger equations

TL;DR

This paper constructs and analyzes multi-bubble blow-up solutions for rough nonlinear Schrödinger equations, revealing their asymptotic behavior, uniqueness conditions, and effects of noise, in both stochastic and deterministic settings.

Contribution

It introduces the first construction of multi-bubble blow-up solutions in rough NLS equations and establishes their conditional uniqueness, extending previous deterministic results to stochastic cases.

Findings

Existence of multi-bubble blow-up solutions in rough NLS equations.

Asymptotic behavior characterized by pseudo-conformal blow-up solutions.

Conditional uniqueness under specific asymptotic conditions.

Abstract

We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schr\"odinger equations in the focusing mass-critical case. In both dimensions one and two, we construct the finite time multi-bubble solutions, which concentrate at $K$ distinct points, $1\leq K<\infty$, and behave asymptotically like a sum of pseudo-conformal blow-up solutions in the pseudo-conformal space $\Sigma$ near the blow-up time. The upper bound of the asymptotic behavior is closely related to the flatness of noise at blow-up points. Moreover, we prove the conditional uniqueness of multi-bubble solutions in the case where the asymptotic behavior in the energy space $H^1$ is of the order $(T-t)^{3+\zeta}$, $\zeta>0$. These results are also obtained for nonlinear Schr\"odinger equations with lower order perturbations, particularly, in the absence of the classical pseudo-conformal symmetry and the conversation law of energy. The existence results are applicable to the canonical deterministic nonlinear Schr\"odinger equation and complement the previous work [43]. The conditional uniqueness results are new in both the stochastic and deterministic case.