二维不可压热传导黏性流体边界层的数学分析

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

本文总结我们对二维不可压热传导黏性流体的运动在小黏性和小热传导极限下的边界层的数学理论分析.在黏性系数和热传导系数为同阶小量的假设下,首先利用多尺度方法得到二维不可压热传导黏性流场的速度边界层和温度边界层所满足的方程;其次在边界层切向速度场关于法向变量单调的假设下,通过运用Crocco变换及能量方法得到此边界层方程在有限阶光滑函数类中的局部适定性;在速度场没有单调性假设的情形下,利用Littlewood-Paley理论建立此边界层方程在关于切向变量解析的函数类中的局部适定性;最后对于一类非单调的解析初值,利用Lyapunov泛函方法得到此边界层方程的解在有限时间内发生爆破的结果,这说明前面得到的解析解一般只能是局部存在的.

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-dimensional incompressible 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.