Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. (August 2016)

• Record Type: Journal Article
• Authors: Dong, Bo-Qing; Jia, Yan
• Is Part Of: Nonlinear analysis. Volume 30(2016)
• Page Start: 41
• Page End: 58
• Publication Date: 2016-08
• Subjects: Navier–Stokes equations -- Stability behaviors -- Energy equality -- Weak–strong uniqueness; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.10.011
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.