Abstract:

Thin layer equations play a crucial role in describing fluid behavior in narrow domains. Typical examples include the Prandtl boundary layer equations and the hydrostatic Euler equations. The former was first proposed by Ludwig Prandtl in 1904 at the Third International Mathematics Congress to provide an effective approximation of solutions to the incompressible Navier-Stokes equations near the spatial boundary. The latter was formally derived in Pierre-Louis Lions’ monograph to describe the leading-order behavior of solutions to the incompressible Euler equations in very thin tubes. In this talk, we will discuss the well-posedness and ill-posedness of these thin layer equations in various contexts, including the latest results concerning steady-state solutions to the hydrostatic Euler equations in nozzles. The primary challenge in solving these equations arises from the potential loss of regularity in the tangential direction. Various mathematical techniques have been developed to address this challenge, and some of these techniques will also be discussed.
This talk includes results from collaborations with Igor Kukavica, Wang Shing Leung, Nader Masmoudi, Robert M. Strain, Vlad Vicol, and Chunjing Xie.