On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory (Journal of Partial Differential Equations, 2020)
The equation arising from Prandtl boundary layer theory $$\frac{\partial u}{\partial t} -\frac{\partial }{\partial x_i}\left( a(u,x,t)\frac{\partial u}{\partial x_i}\right)-f_i(x)D_iu+c(x,t)u=g(x,t)$$ is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since $a(·,x,t)$ may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kružkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if $a_{x_i}(·,x,t)\mid_{x\in \partial Ω}=a(·,x,t)\mid_{x\in \partial Ω}=0$ and $f_i(x)\mid_{x\in \partial Ω}=0$, the stability can be proved even without any boundary value condition.
Keywords: Prandtl boundary layer theory; entropy solution; Kružkov’s bi-variables method; partial boundary value condition; stability.