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One component regularity for the Navier-Stokes equations. (English) Zbl 1149.35069
The authors show that regularity assumptions on one component of the velocity imply regularity of the full solution. These suffficient conditions are certain space-time regularity of the gradient of one component or the component itself. These conditions improve some other one component regularity results, e.g., in [J. Neustupa, A. Novotný and P. Penel, Topics in mathematical fluid mechanics. Meeting on the occasion of Professor John G. Heywood sixtieth birthday, Capo Miseno, Italy, May 27–30, 2000. Rome: Aracne. Quad. Mat. 10, 163–183 (2002; Zbl 1050.35073) and Applied nonlinear analysis. In honor of the 70th birthday of Professor Jindrich Nec̆as. New York, NY: Kluwer Academic/Plenum Publishers, 391–402 (1999; Zbl 0953.35113)].
MSC:
35Q30Stokes and Navier-Stokes equations
35B65Smoothness and regularity of solutions of PDE
76D03Existence, uniqueness, and regularity theory
76D05Navier-Stokes equations (fluid dynamics)
35K15Second order parabolic equations, initial value problems