Year: 2020
Author: Miao Ouyang
Journal of Partial Differential Equations, Vol. 33 (2020), Iss. 2 : pp. 119–142
Abstract
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 Kružkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if $a_{x_i}(\cdot,x,t)\mid_{x\in \partial \Omega}=a(\cdot,x,t)\mid_{x\in \partial \Omega}=0$ and $f_i(x)\mid_{x\in \partial \Omega}=0$, the stability can be proved even without any boundary value condition.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v33.n2.3
Journal of Partial Differential Equations, Vol. 33 (2020), Iss. 2 : pp. 119–142
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Prandtl boundary layer theory entropy solution Kružkov’s bi-variables method partial boundary value condition stability.