Stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains under Navier boundary conditions

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• 作者：W.M. Zaja̧czkowski^a ; ^b ; ^{wz@impan.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Incompressible Navier&ndash ; Stokes equations ; Stability of two-dimensional solutions ; Global regular solutions ; Navier boundary conditions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：444
• 期：1
• 页码：275-297
• 全文大小：472 K

文摘

The Navier–Stokes motions in a cylindrical domain with Navier boundary conditions are considered. First the existence of global regular two-dimensional solutions is proved. The solutions are such that norms bounded with respect to time are controlled by the same constant for all t∈R₊$t\in {\mathbb{R}}_{+}$. Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of a two-dimensional solution, we prove existence of global three-dimensional solutions which remain close to the two-dimensional solution for all time. In this sense we have stability of two-dimensional solutions. Thanks to the Navier boundary conditions the nonlinear term in the two-dimensional Navier–Stokes equations does not influence the energy estimate. This implies that the global two-dimensional solution is proved without any structural restrictions on the external force, initial data or viscosity.