Nonlinear Partial Differential Equations from Continuum and Fluid Mechanics

DMS-9800888 PI：Sideris主要研究者将专注于连续介质和流体力学中出现的偏微分方程的几个代表性问题。光在各向异性晶体中传播的偏微分方程的线性理论将朝着对非线性介质研究有用的方向发展。理论问题需要重新解释和阐述一些现有的技术。将研究与快速重力波相互作用的地球物理流的运动方程。旋转浅水波的准地转极限将与运动方程的重新标度版本的解的寿命相联系。与经典的可压缩流体流动的连接将被强调。将研究平面涡量的增长和传播问题。最终的目标是更好地理解涡旋波与声波的相互作用。当强光，如激光束，穿过晶体时，介质的光学性质随着强度的变化而变化。这些非线性光学现象的数学框架将检查与理解的阈值强度的非线性效应是重要的最终目标。地球物理流动涉及风和地球、海洋和大气层的大尺度动力学。将评估地球引力对这种流动的影响。此外，涡结构可以在流体流动中传播的方式将被检查。

DMS-9800888PI: SiderisThe principal investigator will focus on several representative problems connected with partial differential equations arising in continuum and fluid mechanics.The linear theory of the partial differential equations for light propagation in anisotropic crystals will be developed in a direction that will be useful for the study of nonlinear media. The theoretical problem requires the re-interpretation and elaboration of some of the existing techniques. The equations of motion for geophysical flows interacting with fast gravity waves will be investigated. The quasigeostrophic limit of rotating shallow water waves will be linked to the life span of solutions to a rescaled version of the equation of motion. Connections with classical compressible fluid flow will be emphasized. The problem of the growth and propagation of planar vorticity will be studied. The eventual goal is to achieve a better understanding ofthe interaction of vorticity waves with acoustical waves.When intense light, such as a laser beam, passes through a crystal, the optical properties of the medium change with the intensity. The mathematical framework for these nonlinear optical phenomena will be examined with the eventual goal of understanding the threshold intensity for which nonlinear effects are important. Geophysical flows involve the large-scale dynamics of the winds and currents of the earth oceans and atmospheres. The influence of the earth's gravity on such flows will be assessed. Additionally, the manner in which vortical structures can spread within fluid flow will be examined.