Blowup rate for rotational NLS with a repulsive potential
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2025
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| author | Hu, Yi Lee, Yongki Zheng, Shijun |
| author_facet | Hu, Yi Lee, Yongki Zheng, Shijun |
| contents | In this paper we give an analytical proof of the ``$\log$-$\log$'' blowup rate for mass-critical nonlinear Schrödinger equation (NLS) with a rotation ($Ω\neq 0$) and a repulsive harmonic potential $V_γ(x) = \textrm{sgn}(γ) γ^2 |x|^2$, $γ< 0$ 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 $\mathcal{R}_γ$-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 $|γ|$ for the repulsive potential $V_γ$ can give rise to global in time solution for the focusing RNLS, which is in contrast to the case where $γ$ is positive. This kind of phenomenon was earlier observed in the non-rotational case $Ω= 0$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blowup rate for rotational NLS with a repulsive potential Hu, Yi Lee, Yongki Zheng, Shijun Analysis of PDEs Mathematical Physics In this paper we give an analytical proof of the ``$\log$-$\log$'' blowup rate for mass-critical nonlinear Schrödinger equation (NLS) with a rotation ($Ω\neq 0$) and a repulsive harmonic potential $V_γ(x) = \textrm{sgn}(γ) γ^2 |x|^2$, $γ< 0$ 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 $\mathcal{R}_γ$-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 $|γ|$ for the repulsive potential $V_γ$ can give rise to global in time solution for the focusing RNLS, which is in contrast to the case where $γ$ is positive. This kind of phenomenon was earlier observed in the non-rotational case $Ω= 0$ 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. |
| title | Blowup rate for rotational NLS with a repulsive potential |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2512.22821 |