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

这项研究将研究边界层、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式的数学模型的特解，并在各种科学学科中被广泛使用。例如，它们在生物学、工程学和物理学中具有基本的重要性。利用边界层理论，工程师能够大大简化对固体(如船舶或飞机)附近流体流动的分析。事实证明，这在空气动力学中特别有用。PI将开发数学工具来研究粘性流动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层次的了解。PI还将建立数学标准，在该标准下，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。寻找稳定性标准在实践中很重要，例如，在帮助工程师设计稳定的设备时，否则这些设备可能会被不稳定的海浪破坏。研究集中在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上。要考虑的数学方程包括不可压缩的Euler方程和Navier-Stokes方程，一般的反应扩散方程，以及相对论的Vlasov-Maxwell方程。分析表明，对于足够大的雷诺数，Navier-Stokes方程的一般边界层在频谱上是不稳定的。PI将在傅立叶-拉普拉斯变换方法和Evans函数技术的基础上发展一种非线性理论，以证明边界层近似在零粘性极限下的有效性。这项研究还将提供对光谱稳定源缺陷附近的非线性解的完整动力学的理解。PI将基于空间动力学技术和逐点一维格林函数方法，开发一种新的非线性迭代格式来研究时间周期行波解的稳定性。最后，PI将启动对磁机制的新研究，以限制由相对论Vlasov-Maxwell系统模拟的等离子体。

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.