Long time existence of regular solutions to non-homogeneous Navier-Stokes equations

TL;DR

This paper proves the long-term existence of regular solutions for non-homogeneous Navier-Stokes equations in a bounded cylinder with slip boundary conditions, under small derivative assumptions, using a priori estimates and fixed point theorem.

Contribution

It establishes the global-in-time regularity of solutions for non-homogeneous Navier-Stokes equations with specific boundary conditions, a significant extension in fluid dynamics theory.

Findings

Proved large time regular solutions exist under small derivative conditions.

Used a priori estimates and Leray-Schauder fixed point theorem.

No restriction on the existence time for solutions.

Abstract

We consider the motion of incompressible viscous non-homogeneous fluid described by the Navier-Stokes equations in a bounded cylinder under boundary slip conditions. Assume that the third co-ordinate axis is the axis of the cylinder. Assuming that the derivatives of density, velocity, external force with respect to the third co-ordinate are sufficiently small in some norms we prove large time regular solutions without any restriction on the existence time. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point theorem.