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Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\Bbb R^3$. (English) Zbl 1091.35064
The authors give conditions on vorticity guaranteeing the smoothness of weak solutions to the Navier-Stokes system in $\Bbb R^3$. 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 $\Bbb R^3 \times (0,T)$ with solenoidal initial value $u_0 \in H^1(\Bbb R^3)$. 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 $\frac{2}{q} + \frac{3}{r} = 2$; $\frac{3}{2} < r \leq \infty$, $\sigma \leq \frac{2r}{3}$.

35Q30Stokes and Navier-Stokes equations
76D03Existence, uniqueness, and regularity theory
76D05Navier-Stokes equations (fluid dynamics)
35B65Smoothness and regularity of solutions of PDE
Full Text: DOI
[1] Da Veiga, H. Beirão: A new regularity class for the Navier -- Stokes equations in rn. Chinese ann. Math. 16B, 407-412 (1995) · Zbl 0837.35111
[2] H. Beirão da Veiga, Concerning the regularity problem for the solutions of the Navier -- Stokes equations, C. R. Acad. Sci. Paris, t.321. Série I (1995) 405 -- 408. · Zbl 0840.35075
[3] Da Veiga, H. Beirão: A new regularity class for the L$\infty $(0,T;L3) solutions of the 3-D Navier -- Stokes equations. Port. math. 54, 381-391 (1997)
[4] Bergh, J.; Löfstrom, J.: Interpolation spaces, an introduction. (1976)
[5] Chae, D.; Choe, H. J.: Regularity of solutions to the Navier -- Stokes equations. Electron. J. Differential equations 1999, 1-7 (1999) · Zbl 0923.35117
[6] J.Y. Chemin, Perfect incompressible fluids, Oxford Lectures Series in Mathematics and its Applications, No. 14.
[7] Constantin, P.; Feffermann, C.: Direction of vorticity and the problem of global regularity for the Navier -- Stokes equations. Indiana univ. Math. J. 42, 775-789 (1993) · Zbl 0837.35113
[8] Fabes, E. B.; Jones, B. F.; Riviere, N. M.: The initial value problem for the Navier -- Stokes equations with data in lp. Arch. ration. Mech. anal. 45, 222-240 (1972) · Zbl 0254.35097
[9] Fujita, H.; Kato, T.: On the Navier -- Stokes initial value problem I. Arch. ration. Mech. anal. 16, 269-315 (1964) · Zbl 0126.42301
[10] Giga, Y.: Solutions for semilinear parabolic equations in lp and regularity of weak solutions of the Navier -- Stokes system. J. differential equations 62, 186-212 (1986) · Zbl 0577.35058
[11] Hishida, T.; Izumida, Ken-Ichi: Remarks on a regularity criterion for weak solutions to the Navier -- Stokes equations in rn. Analysis 20, 191-200 (2000) · Zbl 0974.35087
[12] Hopf, E.: Ueber die anfangswertaufgbe für die hydrodynamischen grundgleichungen. Math. nachr. 4, 213-231 (1951) · Zbl 0042.10604
[13] Kozono, H.; Ogawa, T.; Taniuchi, Y.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242, 251-278 (2002) · Zbl 1055.35087
[14] Kozono, H.; Sohr, H.: Regularity criterion on weak solutions to the Navier -- Stokes equations. Adv. differential equations 2, 535-554 (1997) · Zbl 1023.35523
[15] Kozono, H.; Taniuchi, Y.: Bilinear estimates in BMO and the Navier -- Stokes equations. Math. Z. 235, 173-194 (2000) · Zbl 0970.35099
[16] Kozono, H.; Yatsu, N.: Extension criterion via two-components of vorticity on strong solution to the 3 D Navier -- Stokes equations. Math. Z. 246, 55-68 (2003) · Zbl 1060.35105
[17] Leray, J.: Sur le mouvement d’un liquids visqeux emplissant l’espace. Acta math. 63, 193-248 (1934)
[18] Majda, A. J.; Bertozzi, A. L.: Vorticity and incompressible flow. (2002) · Zbl 0983.76001
[19] Masuda, K.: Weak solutions of Navier -- Stokes equations. Tohoku math. J. 36, 623-646 (1984) · Zbl 0568.35077
[20] Serrin, J.: On the interior regularity of weak solutions of the Navier -- Stokes equations. Arch. ration. Mech. anal. 9, 187-195 (1962) · Zbl 0106.18302
[21] Serrin, J.: The initial value problem for the Navier -- Stokes equations. Nonlinear problems, 69-98 (1963) · Zbl 0115.08502
[22] Sohr, H.: Zur regularitätstheorie der instationären gleichungen von Navier -- Stokes. Math. Z. 184, 359-375 (1983) · Zbl 0506.35084
[23] Stein, E. M.: Singular integrals and differentiability properties of functions. (1971)
[24] Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. (1993) · Zbl 0821.42001
[25] Struwe, M.: On partial regularity results for the Navier -- Stokes equations. Commun. pure appl. Math. 41, 437-458 (1988) · Zbl 0632.76034
[26] Takahashi, S.: On interior regularity criteria for weak solutions of the Navier -- Stokes equations. Manuscripta math. 69, 237-254 (1990) · Zbl 0718.35022
[27] Temam, R.: Navier -- Stokes equations. (1977) · Zbl 0383.35057
[28] H. Triebel, Theory of Function Spaces, Monograph in Mathematics, vol. 78, Birkhauser, Basel, 1983.