Nonlinear Partial Differential Equations from Continuum and Fluid Mechanics

DMS-9800888PI：Sideris主要研究与连续介质和流体力学中的偏微分方程组有关的几个典型问题。光在各向异性晶体中传输的偏微分方程组的线性理论将朝着对非线性介质研究有用的方向发展。这个理论问题需要对现有的一些技术进行重新解释和阐述。将研究地球物理流与快速重力波相互作用的运动方程。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联系在一起。将强调与经典可压缩流体流动的联系。将研究平面涡度的增长和传播问题。最终的目标是更好地理解涡度波和声波的相互作用。当强光，如激光穿过晶体时，介质的光学性质会随着强度的变化而变化。我们将研究这些非线性光学现象的数学框架，最终目的是了解对非线性效应很重要的阈值强度。地球物理流动涉及地球、海洋和大气的风和流的大规模动态。我们将评估地球引力对这种流动的影响。此外，还将研究涡旋结构在流体流动中的传播方式。

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.