Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations

TL;DR

This paper introduces a topological method to analyze how large initial velocities influence the long-term behavior of dissipative PDEs invariant under Galilean transformations, demonstrating bounded solutions and attractors.

Contribution

It presents a novel topological approach to study the stabilization effects of large initial velocities in Galilean-invariant dissipative PDEs, including Burgers and Navier-Stokes equations.

Findings

Large initial velocities lead to bounded eternal solutions.

Existence of a locally attracting solution in 3D Navier-Stokes.

Method applies to equations with rapidly oscillating forcing.

Abstract

We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier-Stokes equations we establish existence of a locally attracting solution.