半直线上具有三次非线性的高阶薛定谔方程的较低正则性适定性
Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line
- University of Electronic Science and Technology of China（电子科技大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对半直线上具有三次非线性的高阶薛定谔方程，通过推导三线性估计，在最优正则性下证明了适定性，改进了先前结果并回答了开放问题。
AI中文摘要：
本文继续研究Himonas和Yan关于色散阶数为$2m$且具有三次非线性的非线性薛定谔方程的工作，方程为$iu_t+\left( -1 \right) ^{m+1}\partial _{x}^{2m}u=N_k\left( u,u,u \right)$，其中$m\ge1$为整数，且$N_0\left( u,u,u \right) =uuu$，$N_1\left( u,u,u \right) =\bar{u}uu$，$N_2\left( u,u,u \right) =\bar{u}\bar{u}u$，$N_3\left( u,u,u \right) =\bar{u}\bar{u}\bar{u}$。我们在较低正则性下推导了三线性估计，从而证明了半直线上的cNLS-2m在最优正则性$s=-\frac{m-1}{2}$下是适定的。这改进了之前的结果$s>-\frac{m-1}{2}$，并回答了文献中留下的一个开放问题。此外，对于$k=0,2$，我们也在$s=-\frac{m-1}{2}$处建立了适定性。再者，对于$k=3$，非线性项表现出更强的共振关系，这暗示了在$s>-\frac{2m-1}{3}$时的适定性。我们推导三线性估计依赖于多种Strichartz估计，这与文献中采用的$[k;Z]$-乘子范数方法不同。
英文摘要：
In this paper, we continue the study of Himonas and Yan\cite{himonas2024schrodinger,himonas2026higher,himonas2024higher} on the nonlinear Schrödinger equation with a dispersion of order $2m$ and cubic nonlinearities, \[ iu_t+\left( -1 \right) ^{m+1}\partial _{x}^{2m}u=N_k\left( u,u,u \right), \] where, $m\ge1$ being an integer, \[N_0\left( u,u,u \right) =uuu,\quad N_1\left( u,u,u \right) =\bar{u}uu,\quad N_2\left( u,u,u \right) =\bar{u}\bar{u}u,\quad N_3\left( u,u,u \right) =\bar{u}\bar{u}\bar{u},\] We derive trilinear estimates at lower regularity and thereby prove that the cNLS-2m on the half-line is well-posed at the optimal regularity $s=-\fr{m-1}{2}$. This improves the previous result, $s>-\fr{m-1}{2}$, and answers an open question left in \cite{himonas2026higher}. In addition, for $k=0,2$, we establish well-posedness at $s=-\fr{m-1}{2}$ as well. Moreover, for $k=3$, the nonlinearity exhibits a stronger resonance relation, which implies well-posedness for $s>-\fr{2m-1}{3}$. Our derivation of the trilinear estimates relies on varieties of Strichartz estimates, which differs from the $[k;Z]$-multiplier norm method employed in \cite{himonas2026higher}.
