Stability of the 3D incompressible Navier–Stokes equations with fractional horizontal dissipation

Abstract

The stability problem of Navier–Stokes equations (N–S equations) with fractional dissipation is one of the new areas in Mathematical research. The generalized N–S equations are the equations resulting from replacing −Δ in the N–S equations by (−Δ)α. It has previously been shown that any classical solution of the d-dimensional generalized N–S equations with α≥12+d4 is always global in time. This paper considers the stability problem on the 3D N–S equations with only fractional horizontal dissipation (−Δh)α, where Δh:=∂x12+∂x22. We show that, for any α∈(12,1], the solution corresponding to any sufficiently small initial data in H3(R3) is always global in time and stable in H3(R3). There are many important results on the case when α=1. Our result relaxes this requirement and allows α to go below 1.