C?-Regularity for the Porous Medium Equation.
Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties by using assumptions...
The Special Issue of Kybernetika is devoted to the publication of selected peer-reviewed articles submitted by the participants of the Czech-Japanese Seminar in Applied Mathematics 2008 which took place on September 1-7, 2008 in Takachi-ho and Miyazaki, Japan. The Czech-Japanese Seminar in Applied Mathematics 2008 was organized by the Department of Applied Physics, Faculty of Engineering, University of Miyazaki. It was the fourth meeting in the series of the Czech-Japanese Seminars in Applied Mathematics....
We consider a dense granular shear flow in a two-dimensional system. Granular systems (composed of a large number of macroscopic particles) are far from equilibrium due to inelastic collisions between particles: an external driving is needed to maintain the motion of particles. Theoretical description of driven granular media is especially challenging for dense granular flows. This paper focuses on a gravity-driven dense granular Poiseuille flow...
We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of -coercivity and -growth, for a given parameter . The existence of Dirichlet weak solutions was obtained in [2], in the cases if or if , being the dimension of the domain. In this paper, with help of some new estimates (which lead...