Logarithmically improved blow-up criteria for the 3D nonhomogeneous incompressible Navier-Stokes equations with vacuum. (English) Zbl 1344.35080

Hou, Qianqian; Xu, Xiaojing; Ye, Zhuan

Electron. J. Differ. Equ. 2016, Paper No. 192, 12 p. (2016).

Summary: This article is devoted to the study of the nonhomogeneous incompressible Navier-Stokes equations in space dimension three. By making use of the “weakly nonlinear” energy estimate approach introduced by Z. Lei and Y. Zhou [Discrete Contin. Dyn. Syst. 25, No. 2, 575–583 (2009; Zbl 1171.35452)], we establish two logarithmically improved blow-up criteria of the strong or smooth solutions subject to vacuum for the 3D nonhomogeneous incompressible Navier-Stokes equations in the whole space \(\mathbb R^3\). This results extend recent regularity criterion obtained by H. Kim [SIAM J. Math. Anal. 37, No. 5, 1417–1434 (2006; Zbl 1141.35432)].

MSC:

35Q30	Navier-Stokes equations
35B40	Asymptotic behavior of solutions to PDEs
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05	Navier-Stokes equations for incompressible viscous fluids
35B44	Blow-up in context of PDEs
35D35	Strong solutions to PDEs

Keywords:

nonhomogeneous Navier-Stokes equations; blow-up criterion; strong solution; vacuum