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Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in \(\mathbb R^3\). (English) Zbl 1091.35064
The authors give conditions on vorticity guaranteeing the smoothness of weak solutions to the Navier-Stokes system in \(\mathbb R^3\). Since the pioneering work of Beirao da Veiga, this procedure has been used several times in the work of Kozono, Ogawa, Taniuchi. In a paper of Chae and Choe the conditions of Serrin type are imposed only on two components of vorticity in \(L_q(L_r)\), while a result of Kozono and Yatsu requires conditions on two components of vorticity in \(L_q\)(BMO) and \(r = \infty\). In this paper, an analogous result is proved in homogeneous Besov spaces \(\dot B^0_{r,\sigma}\).
The main result reads as follows: Let \(T > 0\). Suppose \(u(t,x)\) be a weak Leray-Hopf solution to the Navier-Stokes system on \(\mathbb R^3 \times (0,T)\) with solenoidal initial value \(u_0 \in H^1(\mathbb R^3)\). Set \(w = \operatorname{curl} u = [w_1, w_2,w_3], \widetilde{w} = [w_1, w_2, 0] \) and assume that \[ \int_0^T \| \widetilde{w}\|^q_{\dot B^0_{r,\sigma}} \,dt < \infty. \] Then \(u\) is regular provided \(\frac{2}{q} + \frac{3}{r} = 2\); \(\frac{3}{2} < r \leq \infty\), \(\sigma \leq \frac{2r}{3}\).

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