The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations

Abstract

We study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solution $u$ of the Navier-Stokes equations lies in the regular class $\nabla u\in L^p(\mathrm{0},\mathrm{\infty };B_{q,\mathrm{\infty }}^{\mathrm{0}}{(ℝ}^3))$, $\mathrm{(2}\alpha /\mathrm{p)}+\mathrm{(3}/\mathrm{q)}=\mathrm{2}\alpha$, , , then every weak solution $v(x,t)$ of the perturbed system converges asymptotically to $u(x,t)$ as ${∥v\left(t\right)-u\left(t\right)∥}_{L^{\mathrm{2}}}\to \mathrm{0}$, $t\to \mathrm{\infty }$.