Harmonic Analysis and Problems in Mathematical Physics

沈仲伟。DMS-9732894 PI将研究量子领域中出现的几个问题 力学和流体动力学。对于Schr\“odinger算子 电磁势，工作将在估计 负特征值和计数的非经典渐近性 功能其目的是建立有用的估计， 当经典的Cwikel-Lieb-Rosenblum界失效时。 工作也将在基态的半经典分析上完成 能源重点将放在简并势上。 第二条研究路线涉及泡利算子，它描述了 带自旋的带电粒子的运动PI将研究 Pauli算子的Lieb-Thirring型不等式 非均匀的磁场。这种不平等现象， 在研究物质稳定性和半经典 在磁场中分析。最后，工作将继续进行。 粗糙边界区域的边值问题，在许多工程应用中自然出现。 特别是，Stokes方程的预解估计和分数幂 Lipschitz域上的算子。目的是获得 更好地描述了非线性问题的解的行为， 非光滑区域中的Navier-Stokes方程。 这个项目位于数学物理，谐波 分析和偏微分方程。它的目标是更好地了解的光谱性质， 量子系统，并改善数学理论，在此基础上的数值近似和应用， 用于模拟流体流动的Navier-Stokes方程。

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.