某些非线性分数阶偏微分方程的定性研究

The mathematical studies of nonlinear fractional differential equations are one of the hot and difficult issues in the fields of PDEs in resent years. This progect mainly focuses on the existence and the well-posedness of the solution and the existence of the standing wave solution and the travelling wave solution of the nonlinear fractional Schrodinger equations and the nonlinear fractional Landau-Lifshitz equations..Firstly we concentrate on the well-posedness of the nonlinear fractional Schrodinger equations, using the Bourgain space and the method of harmonic analysis to prove the local and global well-posedness of the solution to the fractional Schrodinger equations. Secondly, using the calculus of variations, we obtain the existence and the stability of standing wave solution for the fractional Schrodigner equations. Thirdly, we get the weak and strong sollution of the fractional Landau-Lifshitz equations in high dimensional spaces applying the energy method、Leray-Schauder theory and Viscous elimination method. Finally, by Strichartz estimates and comparison principle, we have the existence and stability of the travelling wave solution of the nonlinear fractional Landau-Lifshitz equations.These problems originate from condensed matter physics and magnetic materials, and they have strong physical background and practical applied value.

非线性分数阶偏微分方程的数学理论研究是近年来偏微分方程研究领域的热点和难点问题之一。本项目主要围绕非线性分数阶Schrodinger方程组和非线性分数阶Landau-Lifshitz方程组解的存在性、适定性及驻波解、行波解的存在性问题展开。.首先研究高维非线性分数阶Schrodinger方程组解的适定性, 应用Bourgain空间和调和分析理论证明解的局部和整体适定性，应用变分方法证明分数阶Schrodinger方程驻波解的存在性及稳定性。接着应用能量方法、 Leray-Schauder原理和粘性消去法研究高维非线性Landau-Lifshitz方程弱解、强解的存在性。最后应用Strichartz估计结合比较原理证明分数阶Landau-Lifshitz方程行波解的存在性和稳定性。.我们所研究问题来源于凝聚态物理和磁性材料学, 具有很强物理背景和实际应用价值。

非线性分数阶偏微分方程的数学理论研究是近年来偏微分方程研究领域的热点和难点问题.之一。本项目主要围绕非线性分数阶Schrodinger方程组和非线性分数阶Landau-Lifshitz方程.组解的存在性、适定性及驻波解、行波解的存在性问题展开。.首先研究高维非线性分数阶Schrodinger方程组解的适定性, 应用Bourgain空间和调和分.析理论证明解的局部和整体适定性，应用变分方法证明分数阶Schrodinger方程驻波解的存在.性及稳定性。接着应用能量方法、 Leray-Schauder原理和粘性消去法研究高维非线性Landau.-Lifshitz方程弱解、强解的存在性。最后应用Strichartz估计结合比较原理证明分数阶Landa.u-Lifshitz方程行波解的存在性和稳定性。.我们所研究问题来源于凝聚态物理和磁性材料学, 具有很强物理背景和实际应用价值。