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Navier-Stokes equations with regularity in one direction. (English) Zbl 1144.81373
Summary: We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity \(\partial u/\partial x_3\) belongs to the space \(L_t^{s_0}L_x^{r_0}\), where \(2/s_0+3/r_0\leq 2\) and \(9/4\leq r_0\leq 3\), then the solution is regular. This extends a result of H. Beirão da Veiga [Chin. Ann. Math., Ser. B 16, No. 4, 407–412 (1995; Zbl 0837.35111)] and [C. R. Acad. Sci. Paris 321, 405–408 (1995; Zbl 0840.35075)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative \(\partial u/\partial x_3\) can be substituted with any directional derivative of \(u\).

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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