Stability and Exponential Decay for the 2D Anisotropic Navier-Stokes Equations with Horizontal Dissipation

Abstract

Solutions to the 2D Navier-Stokes equations with full dissipation in the whole space R-2 always decay to zero in Sobolev spaces. In particular, any perturbation near the trivial solution is always asymptotically stable. In contrast, solutions to the 2D Euler equations for inviscid flows can grow rather rapidly. An intermediate situation is when the dissipation is anisotropic and only one-directional. The stability and large-time behavior problem for the 2D Navier-Stokes equations with only one-directional dissipation is not well-understood. When the spatial domain is the whole space R-2, this problem is widely open. This paper solves this problem when the domain is T x R with T being a 1D periodic box. The idea here is to decompose the velocity u into its horizontal average (u) over tilde and the corresponding oscillation (u) over tilde. By making use of special properties of (u) over tilde, we establish a uniform upper bound and the stability of u in the Sobolev space H-2, and show that (u) over tilde in H-1 decays to zero exponentially in time.