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Volume 33, Issue 2
On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory

Miao Ouyang

DOI: 10.4208/jpde.v33.n2.3

J. Part. Diff. Eq., 33 (2020), pp. 119-142.

Published online: 2020-05

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  • 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.

  • Keywords

Prandtl boundary layer theory, entropy solution, Kružkov’s bi-variables method, partial boundary value condition, stability.

  • AMS Subject Headings

35K65, 35L65, 35R35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mouyang@xmut.edu.cn (Miao Ouyang)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-33-119, author = {Ouyang , Miao}, title = {On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {33}, number = {2}, pages = {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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v33.n2.3}, url = {http://global-sci.org/intro/article_detail/jpde/16855.html} }
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 -

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.

Ouyang , Miao. (2020). On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory. Journal of Partial Differential Equations. 33 (2). 119-142. doi:10.4208/jpde.v33.n2.3
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