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