On the multi-bubble blow-up solutions to rough nonlinear Schrodinger equations

We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schrodinger 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, $1leq K<infty$, 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 Schrodinger 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 Schrodinger equation and complement the previous work [43]. The conditional uniqueness results are new in both the stochastic and deterministic case.