Asymptotic Stability of Landau Solutions to Navier–Stokes System

Abstract

It is known that the three dimensional Navier-Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions, which are axisymmetric and homogeneous of degree −1. We show that these solutions are asymptotically stable under any L 2 -perturbation. (2000): 76D07, 76D05, 35Q30, 35B40. Mathematics Subject Classification

L2− asymptotic stability of the mild solution to the 3D MHD equation

“…On the topic of asymptotic stability for the solutions, there have been many classical results to the Navier-Stokes equation [1,[4][5][6][7][8]10,[12][13][14]. The energy decay problem of weak solutions to the Navier-Stokes equation was originally suggested by Leray in his pioneering papers [9,10].…”

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

“…In particular, we classified in [18] all such solutions with no swirl. (−1)-homogeneous solutions of (1) and (2) have been studied in [3], [9], [13], [14], [15], [19], [22], [23], [24], [25], [28], [29], [30], [31], [35] and [37].…”

Global stability for the fractional Navier‐Stokes equations in the Fourier‐Herz space

“…Auscher et al 12 showed that the global solutions with initial data in VMO −1 are stable. Karch et al 13 have established an asymptotic stability for the global small mild solution under arbitrarily large initial L 2 perturbations. More detailed information on Navier-Stokes equations can be seen in Li and Zheng, 14 Cannone,15 Lemarié-Rieusset, 16 and Temam.…”

“…Using Galerkin method and a modified Fourier splitting technique (see Ogawa et al 22 ), Karch et al 13 have established asymptotic stability for global solution of Navier-Stokes equations under arbitrary large L 2 perturbations. Inspired by this result, we are going to show the stability for global mild solution of system (1) in a critical Fourier-Herz space.…”

“…We state the main results as follows. 13 to the supercritical cases and Fourier-Herz space. In addition, it may be fascinating to establish an asymptotic stability of mild solution to the fractional Navier-Stokes system, which will be our later work.…”