Remarks on Regularity Criteria for Axially Symmetric Weak Solutions to the Navier-Stokes Equations, II

We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of velocity satisfies a weighted Serrin condition and in addition angular component satisfies some condition, then the solution is regular

Nonstationary flow for the Navier-Stokes equations in a cylindrical pipe

In cylindrical domain, we consider the nonstationary flow with prescribed inflow and outflow, modelled with Navier-Stokes equations under the slip boundary conditions. Using smallness of some derivatives of inflow function, external force and initial velocity of the flow, but with no smallness restrictions on the inflow, initial velocity neither force, we prove existence of solutions in $W^{2,1}_2.

Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \le C_*{(r^2 -t)^{-1/2}}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large, we then prove that $v$ is regular at time zero.Comment: 25 page

Global weak solvability of a regularized system of the Navier-Stokes equations for compressible fluid

summary:The paper contains the proof of global existence of weak solutions to the mixed initial-boundary value problem for a certain modification of a system of equations of motion of viscous compressible fluid. The modification is based on an application of an operator of regularization to some terms appearing in the system of equations and it does not contradict the laws of fluid mechanics. It is assumed that pressure is a known function of density. The method of discretization in time is used and finally, a so called energy inequality is derived. The inequality is independent on the regularization used