Mathematical Sciences: Nonlinear Partial Differential Equations

本研究将探讨非线性偏微分方程中的几个问题。用于模拟随时间变化的三维不可压缩流体流动的涡旋方法将得到扩展。非线性演化的模型问题，这是线性不适定的，但仍然有可预测的行为，将研究。本文将研究表示斜相互作用的孤立波或余弦水波的三维水波方程的精确解的存在性。这项研究将扩展对偏微分方程的理论理解，并推进偏微分方程数值解的方法。

This research will examine several problems in nonlinear partial differential equations. Vortex methods for simulating time-dependent, three-dimensional, incompressible fluid flows will be extended. Model problems of nonlinear evolution, which are linearly ill-posed but still have predictable behavior, will be studied. The existence of exact solutions of the three- dimensional water wave equations representing obliquely interacting solitary waves or cnoidal water waves will be investigated. This research will extend theoretical understanding of partial differential equations and advance methodology for their numerical solution.