TY - JOUR
T1 - On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory
AU - Ouyang , Miao
JO - Journal of Partial Differential Equations
VL - 2
SP - 119
EP - 142
PY - 2020
DA - 2020/05
SN - 33
DO - http://doi.org/10.4208/jpde.v33.n2.3
UR - https://global-sci.org/intro/article_detail/jpde/16855.html
KW - Prandtl boundary layer theory, entropy solution, Kružkov’s bi-variables method, partial boundary value condition, stability.
AB - <p style="text-align: justify;">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 $a(\cdot,x,t)$ 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 Kru<span style="line-height: 18.4px; font-family: ">ž</span>kov&#39;s bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if $a_{x_i}(\cdot,x,t)\mid_{x\in \partial \Omega}=a(\cdot,x,t)\mid_{x\in \partial \Omega}=0$ and $f_i(x)\mid_{x\in \partial \Omega}=0$, the stability can be proved even without any boundary value condition.</p>