On the blowup criteria and global regularity for the non-diffusive Boussinesq equations with temperature-dependent viscosity coefficient. (October 2016)

• Record Type: Journal Article
• Authors: He, Ming
• Is Part Of: Nonlinear analysis. Volume 144(2016)
• Page Start: 93
• Page End: 109
• Publication Date: 2016-10
• Subjects: 35Q35 -- 76B03 -- 76D03 -- 76D09; Boussinesq equations -- Temperature-dependent viscosity -- Blowup criterion -- Global regularity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.06.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we consider the Boussinesq equations with temperature-dependent viscosity and zero thermal diffusivity. A blowup criterion of classical solution, depending only on ‖ ∇ μ ( θ ) ‖ L q ( 0, T ; L p ) with p > 2 and 1 / p + 1 / q ≤ 1 / 2, is obtained for the two-dimensional equations. This is in particular consistent with the results in Chae (2006), Hou and Li (2005), Lai et al. (2011), where the global regularity of the 2D non-diffusive Boussinesq equations with constant viscosity coefficient (i.e., μ ( θ ) = Const. ) was proved. A Serrin's type blowup criterion, depending on ‖ u ‖ L s ( 0, T ; L r ) + ‖ ∇ μ ( θ ) ‖ L q ( 0, T ; L p ) with r, p > 3, 3 / r + 2 / s ≤ 1 and 3 / p + 2 / q ≤ 1, is obtained for the three-dimensional case. We also study the global regularity with large data for the two-dimensional equations in a non-divergence form.