AI 中文总结
本文通过计算机辅助的打靶法，证明了三维、四维和五维中能量超临界散焦非线性薛定谔方程存在有限时间爆破解，从而解决了 Bourgain 的猜想。
AI 中文摘要
我们证明了在 $\mathbb R^d$ 中，对于 $(d,p)\in\{(3,27),(4,9),(5,7)\}$，能量超临界散焦非线性薛定谔方程 $i\partial_tu+\Delta u-|u|^{p-1}u=0$ 从光滑、径向、紧支撑的初始数据出发会发生有限时间爆破。解是渐近自相似的，其 $L^\infty$ 范数以标度速率 $(T-t)^{-1/(p-1)}$ 增长，且其临界 Sobolev 范数 $\dot H^{s_c}$ 发散。这解决了 Bourgain 关于三维和四维散焦能量超临界方程全局适定性和散射的猜想。证明依赖于打靶法，并借助计算机辅助。
英文摘要
We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation $i\partial_tu+Δu-|u|^{p-1}u=0$ in $\mathbb R^d$, for $(d,p)\in\{(3,27),(4,9),(5,7)\}$. The solutions are asymptotically self-similar, their $L^\infty$ norm grows at the scaling rate $(T-t)^{-1/(p-1)}$, and their critical Sobolev norm $\dot H^{s_c}$ diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.
Comments44 pages. Python code for the computer-assisted argument is included as suplementary files
