高维下散焦能量超临界NLS的全局适定性与散射
Global well-posedness and scattering for the defocusing energy supercritical NLS in high dimensions
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中文总结 AI 辅助
针对高维散焦能量超临界非线性薛定谔方程，本文通过建立新的非线性估计改进了临界正则性指标s_c的上界，拓展了全局适定性与散射结果的适用范围。
中文摘要 AI 辅助
我们研究维数d≥5时的散焦能量超临界非线性薛定谔方程i∂_t u + Δu = |u|^p u。Killip-Visan（《Comm. Partial Differential Equations》，2010）与Li-Li（《Siam J. Math. Anal.》，2022）已证明：当s_c := d/2 - 2/p > 1时，任何在临界索伯列夫空间Ḣ_x^{s_c}(ℝ^d)中保持有界的解必为全局解且发生散射。在d≥8维的情形下，他们的结果要求p为偶数，或满足s_c < (d+2-√((d-2)²-16))/4。本文通过建立若干新的非线性估计，将s_c的上界改进为s_c < 1+p，从而覆盖了p处于局部存在性范围内的所有情形。
英文摘要
We consider the defocusing energy-supercritical nonlinear Schrödinger equation $i\partial_{t}u+Δu=|u|^p u$ in dimensions $d\ge5$. Killip-Visan [Comm. Partial Differential Equations, 2010] and Li-Li [Siam J. Math. Anal., 2022] proved that for $s_c:=\frac{d}{2}-\frac{2}{p}>1$, any solution that remains bounded in the critical Sobolev space $\dot H_x^{s_c}(\mathbb{R} ^d)$ must be global and scatter. In dimensions \(d \ge 8\), their results required either that \(p\) be even or that \(s_c < \frac{d+2-\sqrt{(d-2)^2-16}}{4}\). In this paper, we improve the upper bound on \(s_c\) to \(s_c<1+p\) by establishing some new nonlinear estimates. This allows us to cover all cases in which \(p\) lies in the local existence range.
