Harmonic Analysis and Problems in Mathematical Physics

沈忠伟。9732894 PI将研究在量子力学和流体力学领域中出现的几个问题。对于具有电磁势的薛定谔算子，我们将对负本征值的估计和计数函数的非经典渐近性进行研究。其目的是在经典Cwikel-Lieb-Rosenblum界失效的情况下建立有用的估计。我们还将对基态能量进行半经典分析。重点将放在简并势上。第二个研究方向是泡利算符，它描述了带自旋的带电粒子的运动。PI将研究具有非均匀磁场的Pauli算子的Lieb-Thirring类型的不等式。这些不等式在研究物质的稳定性和磁场中的半经典分析中起着重要作用。最后将继续研究具有粗糙边界的区域中的边值问题，这在许多工程应用中是自然出现的。特别地，我们将研究Lipschitz域上Stokes算子的预解估计和分数次幂。目的是为了更好地描述非光滑区域上的非线性Navier-Stokes方程的解的性态。该项目位于数学物理、调和分析和偏微分方程组的交界处。它的目标是更好地理解量子系统的光谱特性，并改进模拟流体流动的Navier-Stokes方程的数学理论，作为数值近似和应用的基础。

Zhongwei Shen. DMS-9732894 The PI will study several problems which arise in the fields of quantum mechanics and fluid dynamics. For Schr\"odinger operators with electro-magnetic potentials, work will be done on the estimates of negative eigenvalues and the non-classical asymptotics of the counting function. The objective is to establish useful estimates in the cases when the classical Cwikel-Lieb-Rosenblum bound fails. Work will also be done on the semi-classical analysis of the ground state energy. The emphasis will be on the degenerate potentials. The second line of research concerns the Pauli operator, which describes the motion of charged particles with spin. The PI will study the Lieb-Thirring type inequalities for the Pauli operator with a non-homogeneous magnetic field. Such inequalities play an important role in the study of stability of matter and semi-classical analysis in a magnetic field. Finally work will be continued on boundary value problems in domains with rough boundaries, which arise naturally in many engineering applications. In particular, the resolvent estimates and fractional powers of the Stokes operator in Lipschitz domains will be studied. The objective is to obtain a better description of the behavior of the solutions to the nonlinear Navier-Stokes equations in nonsmooth domains. This project lies at the interface of mathematical physics, harmonic analysis and partial differential equations. Its goal is to gain better understanding of the spectral properties of quantum systems, and to improve the mathematical theory, upon which the numerical approximations and applications are based, for the Navier-Stokes equations which model the fluid flow.