Journal of Applied Mathematics

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 $\nabla u\in {L}^{p}\left(\mathrm{0},\mathrm{\infty };{B}_{q,\mathrm{\infty }}^{\mathrm{0}}{\left(ℝ}^{3}\right)\right)$, $\mathrm{\left(2}\alpha /\mathrm{p\right)}+\mathrm{\left(3}/\mathrm{q\right)}=\mathrm{2}\alpha$, $\mathrm{2}, $\mathrm{0}<\alpha <\mathrm{1}$, then every weak solution $v\left(x,t\right)$ of the perturbed system converges asymptotically to $u\left(x,t\right)$ as ${∥v\left(t\right)-u\left(t\right)∥}_{{L}^{\mathrm{2}}}\to \mathrm{0}$, $t\to \mathrm{\infty }$.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 321427, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808195

Digital Object Identifier
doi:10.1155/2013/321427

Mathematical Reviews number (MathSciNet)
MR3127446

Zentralblatt MATH identifier
06950617

Citation

Wang, Wen-Juan; Jia, Yan. The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations. J. Appl. Math. 2013 (2013), Article ID 321427, 6 pages. doi:10.1155/2013/321427. https://projecteuclid.org/euclid.jam/1394808195

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