Some stability problems to the Navier-Stokes equations in the periodic case

2016/06/15 by Zajaczkowski, Wojciech M.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1606.04701

Abstract

The Navier-Stokes motions in a box with periodic boundary conditions are considered. First the existence of global regular two-dimensional solutions is proved. The solutions are such that continuous with respect to time norms are controlled by the same constant for all t∈ℝ₊. Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of the two-dimensional solutions we prove existence of global three-dimensional regular solutions which remain close to the two-dimensional solutions for all time. In this way we mean stability of two-dimensional solutions.

Related

• Stability of two-dimensional Navier-Stokes motions in the periodic case
• Stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains under Navier-boundary conditions
• Time-periodic solutions to the Navier--Stokes equations on the whole space including the two-dimensional case
• Stabilizing the Long-time Behavior of the Navier-Stokes Equations and Damped Euler Systems by Fast Oscillating Forces
• On the existence of smooth periodic solutions of 3D homogeneous Navier-Stokes equations