论文标题
stokes and navier-stokes方程在均匀$ c^{2,1} $ - $ l_q $ -spaces中的部分滑移。
Stokes and Navier-Stokes equations subject to partial slip on uniform $C^{2,1}$-domains in $L_q$-spaces
论文作者
论文摘要
本说明涉及均匀$ c^{2,1} $ - $ l_q $上的stokes和navier-stokes方程的良好性。特别是，非螺旋型域的类别，即不存在Helmholtz分解的域。一方面，事实证明，在通常适用于Helmholtz域的标准设置中，通常受部分滑动的stokes方程并不适合。另一方面，事实证明，在某些合理的假设下，受部分滑移的stokes和navier-Stokes方程在广义环境中得到了充分的范围。此设置依赖于Helmholtz分解的广义版，该版本在交叉路口的适当条件下以及$ L_Q $中的梯度和螺线管字段的总和。事实证明，Stokes分解问题的良好性甚至相当于存在普遍的Helmholtz分解。例如，提出的方法包括Bogovski \uı和Maslennikova引入的类似扇区的非螺旋型域，以及其他均匀的均匀$ c^{2,1} $域。
This note concerns well-posedness of the Stokes and Navier-Stokes equations on uniform $C^{2,1}$-domains on $L_q$. In particular, classes of non-Helmholtz domains, i.e., domains for which the Helmholtz decomposition does not exist, are adressed. On the one hand, it is proved that the Stokes equations subject to partial slip in general are not well-posed in the standard setting that usually applies for Helmholtz domains. On the other hand, it is proved that under certain reasonable assumptions the Stokes and Navier-Stokes equations subject to partial slip are well-posed in a generalized setting. This setting relies on a generalized version of the Helmholtz decomposition which exists under suitable conditions on the intersection and the sum of gradient and solenoidal fields in $L_q$. The proved well-posedness of the Stokes resolvent problem turns even out to be equivalent to the existence of the generalized Helmholtz decomposition. The presented approach, for instance, includes the sector-like non-Helmholtz domains introduced by Bogovski\uı and Maslennikova as well as further wide classes of uniform $C^{2,1}$-domains.
