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超临界离焦非线性薛定谔系统的有限时间爆破

Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

Terence Tao

备注：
55 pages, no figures, submitted, Analysis & PDE
发表信息：
Analysis & PDE 11 (2018) 383-438
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摘要：

本文考虑Galilean时空${\bf R} \times {\bf R}^d$上的散焦非线性薛定谔系统$$ i \partial_t + Δu = (\nabla_{{\bf R}^m} F)(u) + G $$的整体正则性问题，其中场$u\colon {\bf R}^{1+d} \to {\bf C}^m$是向量值场，$F\colon {\bf C}^m \to {\bf R}$是一个光滑势，它是正的，相位旋转不变的，并且对于某个指数$p >1$，在单位球外是$p+1$阶齐次的，$G: {\bf R} \times {\bf R}^d \to {\bf C}^m$是一个光滑的紧支撑强迫项.推广了标量散焦非线性薛定谔（NLS）方程，其中$m=1$和$F(v) = \frac{1}{p+1} |v|^{p+1}$ .本文研究了$d \geq 3$和$p > 1 + \frac{4}{d-2}$的超临界情况。我们表明，在这种情况下，存在一个光滑的潜在$F$为一些足够大的$m$，积极和均匀的顺序$p+1$以外的单位球，和一个光滑的companitary选择的初始数据$u(0)$和强制项$G$的解决方案开发一个有限的时间奇异性。事实上，该解是局部离散自相似的关于抛物重新标度的时空。这表明，人们不能希望建立一个整体的正则性结果的标量散焦NLS，除非使用一些特殊的性质，该方程是不共享这些散焦非线性薛定谔系统。正如作者在前一篇论文中考虑的非线性波动方程的类似问题，基本策略是首先选择$u$的质量，动量和能量密度，然后选择$u$本身，最后设计潜在的$F$，以解决所需的方程。

英文摘要：

We consider the global regularity problem for defocusing nonlinear Schrödinger systems $$ i \partial_t + Δu = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$, and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schrödinger (NLS) equation, in which $m=1$ and $F(v) = \frac{1}{p+1} |v|^{p+1}$. In this paper we study the supercritical case where $d \geq 3$ and $p > 1 + \frac{4}{d-2}$. We show that in this case, there exists a smooth potential $F$ for some sufficiently large $m$, positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth compactly choice of initial data $u(0)$ and forcing term $G$ for which the solution develops a finite time singularity. In fact the solution is locally discretely self-similar with respect to parabolic rescaling of spacetime. This demonstrates that one cannot hope to establish a global regularity result for the scalar defocusing NLS unless one uses some special property of that equation that is not shared by these defocusing nonlinear Schrödinger systems. As in a previous paper of the author considering the analogous problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation.

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