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.
