Large time existence of solutions to the Navier-Stokes equations in axially symmetric domains with inflow and outflow
This work is concerned with establishing the long time existence of some regular axisymmetric solutions of Navier-Stokes equations. The incompressible viscous fluid is occupying an axisymmetric finite pipe of varying cross sections. The classical constitutive relations are assumed. On the lateral surface of the domain a slip boundary condition is adopted, i.e., the normal component of the velocity vanishes whereas shear forces are proportional on tangential components of the velocity on that surface. Two end sections are subjected to given inflows and outflows satisfying compatibility condition imposed by incompressibility. Shear forces are supposed to vanish on end sections. Initial velocity field is also given. The aim of the work is to show the long time existence of regular solutions without imposing any restriction on magnitudes of the body force density, initial and boundary data. To this end, the author employs the approach of regularising weak solutions. In order to achieve this aim, the author first introduces some auxiliary functions satisfying certain appropriate field equations and boundary conditions. Then evaluate some long time a priori estimates of certain expressions by utilising properties of various order weighted Sobolev spaces and Lebesgue spaces. The use of them in the Leray-Schauder fixed point theorem then leads to the desired results.
scientific article; zbMATH DE number 5620897