Dynamics of Wave Structures in Fluid Dynamics, Oscillatory Media, and Plasma Physics

研究将调查边界层，相干结构和等离子体平衡的动力学。这些对象被表示为涉及偏微分方程的数学模型的特殊解，并已广泛应用于各个科学学科。例如，它们在生物学、工程学和物理学中具有根本的重要性。利用边界层理论，工程师能够大大简化固体（如船舶或飞机）附近流体流动的分析。这在空气动力学中被证明是非常有用的。PI将开发数学工具来研究粘性流边界层简化的有效性，从而提供对物理观测和实验室实验的更深入理解。PI还将建立数学标准，根据这些标准，振荡介质中的相干结构和等离子体的平衡在扰动下表现良好。寻找稳定性准则在实践中很重要，例如，帮助工程师设计稳定的设备，否则这些设备可能会被不稳定的波浪损坏。研究重点是与流体动力学中的边界层动力学、振荡介质中相干结构的稳定性以及等离子体的磁约束有关的数学问题。要考虑的数学方程包括不可压缩的欧拉方程和Navier-Stokes方程，一般的反应扩散系统，和相对论的Vlasov-Maxwell系统。通用的Navier-Stokes方程的边界层分析表明，足够大的雷诺数是频谱不稳定的。PI将开发一个非线性理论，建立在傅立叶-拉普拉斯变换方法和埃文斯函数技术的基础上，以证明边界层近似在粘性极限消失时的无效性。这项研究还将提供一个完整的动态非线性解决方案附近的光谱稳定源缺陷的理解。PI将开发一种新的非线性迭代方案来研究时间周期行波解的稳定性，基于空间动力学技术和逐点一维绿色函数方法。最后，PI将启动新的研究磁机制，以限制相对论Vlasov-Maxwell系统建模的等离子体。

The research will investigate the dynamics of boundary layers, coherent structures, and plasma equilibria. These objects are expressed as special solutions to mathematical models involving partial differential equations and have been widely used in various scientific disciplines. They are for instance of fundamental importance in biology, engineering, and physics. Utilizing boundary layer theory, engineers are able to significantly simplify the analysis of fluid flows near a solid body, such as a ship, or an airplane. This has proven exceptionally useful in aerodynamics. The PI will develop mathematical tools to study the validity of boundary layer simplifications of viscous flows, and thereby provide a deeper understanding of physical observations and laboratory experiments. The PI will also establish mathematical criteria, under which coherent structures in oscillatory media and equilibria of a plasma are well behaved under disturbances. The search for a stability criterion is important in practice, for instance, in helping engineers design stable devices, which could otherwise be damaged by unstable waves. The research focuses on mathematical questions concerned with the dynamics of boundary layers in fluid dynamics, the stability of coherent structures in oscillatory media, and the magnetic confinement of a plasma. The mathematical equations to be considered include the incompressible Euler and Navier-Stokes equations, general reaction-diffusion systems, and the relativistic Vlasov-Maxwell systems. Generic boundary layers of the Navier-Stokes equations are analytically shown to be spectrally unstable for sufficiently large Reynolds numbers. The PI will develop a nonlinear theory, building on the Fourier-Laplace transformed approach and the Evans function techniques, to prove the invalidity of boundary layer approximations in the vanishing viscosity limit. The research will also provide an understanding of the complete dynamics of nonlinear solutions near spectrally stable source defects. The PI will develop a novel nonlinear iteration scheme to study the stability properties of time-periodic traveling wave solutions, based on the spatial-dynamics techniques and the pointwise one-dimensional Green's function approach. Finally, the PI will initiate new investigations on magnetic mechanisms to confine a plasma modeled by the relativistic Vlasov-Maxwell systems.