Enhanced dissipation for the third component of 3D anisotropic Navier-Stokes equations

Abstract

In this paper, we study the decay rates for the global small smooth solutions to 3D anisotropic incompressible Navier-Stokes equations. In particular, we prove that the horizontal components of the velocity field decay like the solutions of 2D classical Navier-Stokes equations. While the third component of the velocity field decays as the solutions of 3D Navier-Stokes equations. We remark that such enhanced decay rate for the third component is caused by the interplay between the divergence free condition of the velocity field and the horizontal Laplacian in the anisotropic Navier-Stokes equations.

Large time behavior of solutions to the $3$D anisotropic Navier-Stokes equation

“…(1) The decay rate of (1.4)and (1.5) with p = 2 is same as that of the L 2 -norm of the 2D Gaussian, which corresponds to the limiting case σ = 1 in (1.2) and (1.3). Thus, our result extends the previous studies [11,20].…”

“…We refer to [3,6,7,13] for the large time behavior of isotropic Navier-Stokes equations. The purpose of this paper is to refine the results obtained by [11,20] and to clarify the effect of the anisotropy on the large time behavior of the solution in terms of L p decay rates and asymptotic expansions.…”

“…The decay rate of (1.2) coincides to that of the linear solution e t∆ h u 0 . Xu and Zhang [20] relaxed the regularity condition of [11] and showed that for s > 2 and (1 + 3s)/{10(s − 1)} < σ < 1, the solution u of (1.1) with given small initial data u 0 ∈ ( Ḣ0,s ∩ Ḣ−σ,0 ∩ Ḣ−σ,− σ 2 − 1 4 ∩ Ḣ− 1 2 ,1 )(R 3 ) satisfies that for α h ∈ (N ∪ {0}) 2 with |α h | 1, 4 ),…”

3D anisotropic Navier–Stokes equations in T2×R : stability and large-time behaviour

“…The work of Xu and Zhang [28] weakens the regularity assumption of [14] on the initial data by employing anisotropic Besov spaces. They also observed the remarkable enhanced dissipation in the third component of the velocity, which decays faster than the first two components!…”