某些流体动力学方程与非线性色散方程的数学研究

The project intends to study the mathematical theory of some partial differential equations which appear in fluid mechanics and quantum mechanics by developing the modern analysis methods, mainly related to: the research on the mathematical theory of the fluid dynamics equations such as Maxwell-Navier-Stokes equations; the research on scattering theory of nonlinear dispersive equation with potential by developing the harmonic analysis theory associated with Laplacian with potential , combining the argument of concentration compactness/rigidity developed by Kenig-Merle; the research of nonlinear dispersive equations on smooth compact Reimann manifold by studying discrete restriction estimates, wave packet decomposition and the corresponding micro-local analysis, combined with summation of trigonometric series average appearing in additive number theory.

本项目拟通过发展现代分析的方法，研究流体力学与量子力学中出现的某些偏微分方程的数学理论，主要涉及：通过发展物理空间与频率空间的局部化技术研究Maxwell-Navier-Stokes 方程等流体动力学方程；通过发展含位势的Laplace算子对应的调和分析理论、结合Kenig-Merle的集中紧/刚性方法研究含位势的非线性色散方程的散射理论；通过研究离散限制性估计、波包分解及相应的微局部分析，结合堆垒数论中的三角级数的平均求和法，研究光滑紧流形上的非线性色散方程等。

通过发展调和分析的方法，本项目对流体力学与量子力学中出现的某些偏微分方程的数学理论进行了深入的研究,取得一系列突破性的成果。圆满完成了项目的预期目标。例如：通过发展物理空间与频率空间的局部化技术，解决了三维不可压缩分数阶Ｎavier-Stokes方程(5/6＜α≤ 1) 向前自相似解的存在性、衰减估计和整体正则性。通过发展含位势的Laplace算子对应的调和分析理论，解决了具反平方位势能量临界波动方程散射猜想等.