AI 中文总结
研究一维分数阶非线性薛定谔方程在傅里叶 - 索伯列夫空间的适定性，通过证明费弗曼 - 斯坦型估计，利用其不等式及推广，克服导数损失困难，得出大数据局部和小数据全局适定性结果。
AI 中文摘要
本文证明了分数阶薛定谔方程的一个精确的费弗曼 - 斯坦型估计，它可视为傅里叶勒贝格空间\(\widehat{L^p}\)中数据的广义斯特里赫茨估计。作为费弗曼 - 斯坦不等式及其非对角推广的应用，我们证明了一维纯幂次非线性分数阶非线性薛定谔方程在齐次和非齐次傅里叶 - 索伯列夫空间\(\widehat{\dot{H}^s_p},\widehat{H^s_p}\)中的大数据局部适定性和小数据全局适定性结果。通过在\(L_x^r(\mathbb{R};L^q_t(I))\)空间中建立解来克服标准斯特里赫茨估计中导数损失的困难。
英文摘要
In this paper, we prove a sharp Fefferman--Stein-type estimate for the fractional Schrödinger equation, which can be regarded as a generalized Strichartz estimate for data in the Fourier Lebesgue space $\widehat{L^p}$. Then, as an application of the Fefferman--Stein inequality and its off-diagonal generalization, we prove large data local well-posedness and small data global well-posedness results for the one dimensional fractional nonlinear Schrödinger equation with pure power nonlinearities in the homonegeneous and inhomogeneous Fourier--Sobolev spaces $\widehat{\dot{H}^s_p},\widehat{H^s_p}$. Solutions are established in $L_x^r(\mathbb{R} ;L^q_t(I))$ spaces in order to overcome the difficulty of a loss of derivatives in the standard Strichartz estimates.
Comments44pages, 2 figures
