Blowup rate for rotational NLS with a repulsive potential

Abstract

In this paper we give an analytical proof of the ``$\log$-$\log$'' blowup rate for mass-critical nonlinear Schr\"odinger equation (NLS) with a rotation ($\Omega \neq 0$) and a repulsive harmonic potential $V_{\gamma}(x) = \textrm{sgn}(\gamma) \gamma^2 |x|^2$, $\gamma < 0$ when the initial data has a mass slightly above that of $Q$, the ground state solution to the free NLS. The proof is based on a virial identity and an $\mathcal{R}_{\gamma}$-transform, a pseudo-conformal transform in this setting. Further, we obtain a limiting behavior description concerning the mass concentration near blowup time. A remarkable finding is that increasing the value $|\gamma|$ for the repulsive potential $V_{\gamma}$ can give rise to global in time solution for the focusing RNLS, which is in contrast to the case where $\gamma$ is positive. This kind of phenomenon was earlier observed in the non-rotational case $\Omega = 0$ in Carles' work. In addition, we provide numerical simulations to partially illustrate the blowup profile along with the blowup rate using dynamic rescaling and adaptive mesh refinement method.