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Blowup rate for rotational NLS with a repulsive potential

Yi Hu, Yongki Lee, Shijun Zheng

TL;DR

Addressing the mass-critical nonlinear Schrödinger equation with rotation and a repulsive harmonic potential, the paper proves a log-log blowup rate for initial data near the ground state, via a virial identity and the $\mathcal{R}_\gamma$-transform that connects RNLS to the classical NLS. It establishes an $L^2$-threshold framework, analyzes limiting behavior and mass concentration near blowup, and shows that larger $|\gamma|$ can yield global solutions, contrasting with the attractive case. The results are complemented by numerics in 2D that illustrate blowup profiles and rates, supporting the theoretical predictions. Overall, the work extends the log-log blowup theory to rotating, repulsive-oscillator settings and provides a detailed mechanism for mass concentration and lifespan modulation by the repulsive potential.

Abstract

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.

Blowup rate for rotational NLS with a repulsive potential

TL;DR

Addressing the mass-critical nonlinear Schrödinger equation with rotation and a repulsive harmonic potential, the paper proves a log-log blowup rate for initial data near the ground state, via a virial identity and the -transform that connects RNLS to the classical NLS. It establishes an -threshold framework, analyzes limiting behavior and mass concentration near blowup, and shows that larger can yield global solutions, contrasting with the attractive case. The results are complemented by numerics in 2D that illustrate blowup profiles and rates, supporting the theoretical predictions. Overall, the work extends the log-log blowup theory to rotating, repulsive-oscillator settings and provides a detailed mechanism for mass concentration and lifespan modulation by the repulsive potential.

Abstract

In this paper we give an analytical proof of the ``-'' blowup rate for mass-critical nonlinear Schrödinger equation (NLS) with a rotation () and a repulsive harmonic potential , when the initial data has a mass slightly above that of , the ground state solution to the free NLS. The proof is based on a virial identity and an -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 for the repulsive potential can give rise to global in time solution for the focusing RNLS, which is in contrast to the case where is positive. This kind of phenomenon was earlier observed in the non-rotational case 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.
Paper Structure (18 sections, 18 theorems, 124 equations, 4 figures)

This paper contains 18 sections, 18 theorems, 124 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rotational NLS on $\mathbb{R}^n$
  4. Solution to the linear flow of RNLS (\ref{['eq:nls_va']})
  5. Local theory
  6. Threshold for $L^2$-critical focusing RNLS
  7. Blowup criterion
  8. Blowup result if $\left\| u\right\| _2\ge \left\| Q\right\| _2$
  9. Limiting behavior near ground state profile
  10. Spectral property and the $\log$-$\log$ law
  11. $\mathcal{R}_\gamma$-transform for $-\Delta -{\gamma^2}|x|^2 + i A \cdot \nabla$
  12. Pseudo-conform transform
  13. Solutions of RNLS that blowup at infinity time
  14. The $\log$-$\log$ blowup rate for rotational NLS
  15. Limiting behavior: concentration of mass
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $p = 1 + \frac{4}{n}$ and $V_\gamma(x)=-\gamma^2|x|^2$, $\gamma<0$ in equation eq:nls_va, where $1\leq n \leq 10$. Suppose $u_0\in \mathcal{S}_{\alpha*}$, namely, and Then there exists ${T^*} =T_{max}< \infty$ such that $u\in C([0,{T^*}); \mathscr{H}^1)$ is a blowup solution to equation eq:nls_va with the $\log$-$\log$ blowup rate where $Q=Q_0$ is the unique solution to equation eq:ground_s

Figures (4)

  • Figure 1: Solution for $u_0=5\exp(-(2x)^2-y^2)$ at $t\approx 0.10343$ for the repulsive harmonic potential case $\tilde{\gamma}=-1$. Top left: $|u(x,y)|$; top right: zoom for $|u(x,y)|$; bottom left: solution $|u(\xi,\eta)|$ on the computational domain. bottom right: the mesh profile.
  • Figure 2: Solution profile for $u_0=5\exp(-(2x)^2-y^2)$ at $t\approx 0.10343$ for the repulsive harmonic potential case. Left: $|u(t,x,0)|$ and $|u(t,0,y)|$ and rescaled $Q_c$. Right: $L(t)=\|\nabla u \|_2$ on the log scale.
  • Figure 3: Solution for $u_0=5\exp(-(2x)^2-y^2)$ at $t\approx 0.10199$ for the attractive harmonic potential case $\tilde{\gamma}=1$. Top left: $|u(x,y)|$; top right: zoom for $|u(x,y)|$; bottom left: solution $|u(\xi,\eta)|$ on the computational domain. bottom right: the mesh profile.
  • Figure 4: Solution profile for $u_0=5\exp(-(2x)^2-y^2)$ at $t\approx 0.10199$ for the attractive harmonic potential case. Left: $|u(t,x,0)|$ and $|u(t,0,y)|$ and rescaled $Q_c$. Right: $L(t)=\|\nabla u \|_2$ on the log scale.

Theorems & Definitions (37)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • proof : The $\Sigma$-boundedness for $u(t)$ may not be obtained as in the trapping case $\gamma>0$
  • ...and 27 more