On the two-dimensional Navier-Stokes equations with horizontal viscosity

TL;DR

This paper investigates the well-posedness, stability, and long-term behavior of solutions to a 2D channel flow governed by Navier-Stokes equations with horizontal viscosity only, under minimal initial regularity assumptions.

Contribution

It establishes global well-posedness and analyzes the stability of solutions for the 2D Navier-Stokes equations with horizontal viscosity, requiring less initial regularity than previous studies.

Findings

Proves global existence of solutions under minimal regularity.

Analyzes the large-time behavior of solutions.

Studies stability properties of the flow.

Abstract

This paper is concerned with a 2D channel flow that is periodic horizontally but bounded above and below by hard walls. We assume the presence of horizontal viscosity only. We study the well-posedness, large-time behavior, and stability of solutions. For global well-posedness, we aim to assume less differentiability on initial velocity $(u_0, v_0)$: in particular, we assume $u_0,v_0\in L^2(\Omega)$ and $\partial_y u_0 \in L^2(\Omega)$.