非线性分形薛定谔方程和分形Strichartz估计的研究

This project belongs to the interface of fractal geometry and the theory of partial differential equations. To studying nonlinear fractal Schrödinger equations defined by fractal measures, we will establish tools including fractal Strichartz estimates and conservation laws. Precisely, the main research contents of this project are as follows: (1) we prove fractal Strichartz estimates for fractal Shrödinger equations; (2) by using fractal Strichartz estimates and techniques from classical theory of partial differential equations, we study the local well-posedness of nonlinear fractal Schrödinger equations, including the existence and uniqueness of a local solution, blowup alternative, continuous dependence of the initial value; (3) by building conservation laws and energy estimates, we consider global well-posedness of nonlinear fractal Schrödinger equations. The research of this project will not only enrich and develop the theory of fractal geometry and partial differential equations, but also lead to applications in quantum mechanics and scale relativity theory.

本项目的研究内容属于分形几何和偏微分方程等的交叉领域.项目拟通过建立分形Strichartz估计和守恒律等工具,研究由分形测度定义的非线性分形薛定谔方程.具体研究内容如下:(1)建立适用于分形薛定谔方程的分形Strichartz估计; (2)利用分形Strichartz估计和经典偏微分方程理论中的方法,研究非线性薛定谔方程的局部适定性问题,包括局部解的存在性与唯一性,爆破准则和对初值函数的连续依赖性等内容;(3)通过建立守恒律和能量估计,研究非线性分形薛定谔方程的整体适定性问题.本项目的研究不仅将丰富和发展分形几何和偏微分方程等理论, 而且在量子力学和尺度相对论等领域中较强的应用价值.

本项目着重研究了分形上的热方程、波动方程、薛定谔方程、薛定谔算子和Strichartz估计等分析问题。具体研究内容如下：(1) 利用分形测度的几何性质和经典偏微分方程的理论，我们得到了分形上的热方程、波动方程、薛定谔方程解的存在唯一性和数值逼近等结果。(2) 利用分形Laplace算子的分析性质和经典分析中的技巧，得到了分形薛定谔算子的特征值渐进估计等结果。本项目已发表SCIE论文5篇，已接收待发表SCIE论文1篇，已投稿论文2篇。项目负责人参加会议5次, 作学术报告1次。