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Navier-Stokes equations with vorticity in Besov spaces of negative regular indices. (English) Zbl 1339.35221
Summary: This paper studies the Cauchy problem for the three-dimensional Navier-Stokes equations, and shows that the condition \[ \operatorname{\nabla} \times \mathbf{u} \in L^{\frac{2}{2 - r}}(0, T; \dot{B}_{\infty, \infty}^{- r}), 0 < r < 2 \] ensures the regularity of the solution on \((0, T)\). This improves and extends many previous results.

MSC:
35Q30 Navier-Stokes equations
35B65 Smoothness and regularity of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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