Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains
Michal Beneš
Conference Publications 2011, 2011(Special): 135-144 doi: 10.3934/proc.2011.2011.135
We study a mixed initial{boundary value problem for the Navier{ Stokes equations, where the Dirichlet, Neumann and slip boundary conditions are prescribed on the faces of a three-dimensional polyhedral domain. We prove the existence, uniqueness and smoothness of the solution on a time interval (0, $T$*), where 0 $< T$* $<= T$.
keywords: Navier-Stokes equations mixed boundary conditions regularity of generalized solutions
Area preserving geodesic curvature driven flow of closed curves on a surface
Miroslav KolÁŘ Michal BeneŠ Daniel ŠevČoviČ
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3671-3689 doi: 10.3934/dcdsb.2017148

We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.

keywords: Geodesic curvature driven flow surface area preserving flow Hölder smooth solutions flowing finite volume method

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