直线上中间非线性薛定谔方程的次适定性
Sub-critical well-posedness for the intermediate nonlinear Schrödinger equation on the line
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中文总结 AI 辅助
本文研究中间非线性薛定谔方程的适定性，证明其在$s>0$时局部适定、$0<s<1/2$且初始数据$L^2$范数较小时全局适定，方法不依赖方程完全可积性，还得到连续Calogero-Moser方程的对应结果。
中文摘要 AI 辅助
我们继续研究中间非线性薛定谔方程（INLS）的适定性理论。首先，我们证明INLS在任意$s>0$时在$H^s(\boldsymbol{R})$中是局部适定的，这改进了我们之前任意$s>1/4$时的局部适定性结果，覆盖了INLS的全部标度次临界范围。特别地，我们还在无手性假设的全部标度次临界范围内得到了连续Calogero-Moser方程的局部适定性。我们的方法依赖于规范变换、四个辅助变量的闭系统推导以及非线性光滑估计，而非利用这些方程的完全可积性。其次，对于可积模型，我们证明当初始数据的$L^2$范数较小时，在$0<s<1/2$范围内是全局适定的。此外，我们表明只要流保持$L^2$等连续集，我们的全局适定性结果就适用，因此小数据限制可通过先验等连续性结果消除。我们的论证基于我们前期工作中发现的Lax对所构造的新型守恒量族。
英文摘要
We continue our study of the well-posedness theory for the intermediate nonlinear Schrödinger equation (INLS). Firstly, we prove that INLS is locally well-posed in $H^s (\mathbb{R})$ for any $s>0$. This improves on our previous result of local well-posedness for any $s>\frac 14$, and covers the full scaling-subcritical range for INLS. In particular, we also obtain the local well-posedness for the continuum Calogero-Moser equation without chirality assumption in the full scaling-subcritical range. Our method relies on a gauge transformation, the derivation of a closed system for four auxiliary variables, and nonlinear smoothing estimates, but not on the completely integrable nature of these equations. Secondly, for the integrable models, we prove global well-posedness for $0<s<\frac12$ for initial data with small $L^2$-norm. Moreover, we show that our global well-posedness result applies whenever $L^2$-equicontinuous sets are preserved by the flow, and so the small-data restriction would be removed by an a-priori equicontinuity result. Our argument relies on a novel family of conserved quantities based upon the Lax pair we discovered in our prior work.
