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We show that the spatial Lq-norm (q > 5/3) of the vorticity of an incompressible viscous fluid in ℝ3 remains bounded uniformly in time, provided that the direction of vorticity… Click to show full abstract
We show that the spatial Lq-norm (q > 5/3) of the vorticity of an incompressible viscous fluid in ℝ3 remains bounded uniformly in time, provided that the direction of vorticity is Holder continuous in space, and that the space-time Lq-norm of vorticity is finite. The Holder index depends only on q. This serves as a variant of the classical result by Constantin-Fefferman (Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. J. Math. 42 (1993), 775–789), and the related work by Grujic-Ruzmaikina (Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE, Indiana Univ. J. Math. 53 (2004), 1073–1080).
Journal Title: Acta Mathematica Scientia
Year Published: 2020
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