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A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component. (English) Zbl 1166.35359
Summary: We consider the regularity criteria for the weak solutions to the Navier-Stokes equations in \(\mathbb{R}^3\). It is proved that if the gradient of any one component of the velocity field belongs to \(L^{\alpha,\gamma}\) with \(2/\alpha + 3/\gamma = 3/2\), \(3\leq\gamma < \infty\), then the weak solution actually is strong.

35Q30 Navier-Stokes equations
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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