Mathematical analysis of boundary layers in two-dimensional incompressible viscous heat conducting flows

Abstract

This paper is to review our recent study on the well-posedness and blowup of the boundary layer equations in small viscosity and heat conductivity limit for the two-dimensionalincompressible viscous heat conducting flows near a physical boundary. In the case that the viscosity and heat conductivity have the same scale, first we derive the boundary layer equations of the viscous layer and thermal layer for the incompressible Navier-Stokes-Fourier equations by multi-scale analysis,and then we shall review a well-posedness result established under the monotonicity condition of tangential velocity by using the Crocco transformation and the energy method.After that, when the tangential velocity does not satisfy the monotonicity assumption, we shall present a well-posedness result when the data are analytic with respect to the tangential variable,by using the Littlewood-Paley theory. We also present a blowup result in a finite time by introducing a Lyapunov functional, when the monotonicity condition is violated for the initial velocity. This shows that the analytic solution which we obtained exists in a finite time only in general.