## The BKM criterion for the 3D Navier-Stokes equations via two velocity components.(English)Zbl 1196.35153

From the paper: The incompressible fluid motion in the whole space $$\mathbb R^3$$ is governed by the Navier-Stokes equations
$\begin{cases} \partial_tu+(u\cdot\nabla)u+\nabla\pi=\Delta u,\\\nabla\cdot u=0,\\ u(x,0)=u_0.\end{cases}$
Here $$\nabla$$ represents the gradient $$(\partial_1,\partial_2,\partial_3)$$, $$u_0$$ is a given initial velocity, $$u=(u_1,u_2,u_3)$$ and $$\pi$$ denote the unknown velocity vector Field and scalar pressure field of the fluid motion, respectively. Here and in what follows, we use the notations for a vector function $$u$$,
$(u\cdot\nabla)u=\sum^3_{i=1}u_i\partial_i u_k\quad (k=1,2,3),\quad \nabla\cdot u=\sum^3_{i=1}\partial_iu_i.$
In the study of the regularity criterion of Leray-Hopf weak solutions to the 3D Navier-Stokes equations, the Beale-Kato-Majda type criterion is obtained in terms of the horizontal derivatives of the two velocity components
$\int^T_0\|\nabla_h\widetilde u(s)\|_{\dot B^0_{\infty,\infty}}\,ds<\infty,\quad \widetilde u=(u_1,u_2,0),\quad \nabla_h\widetilde u=(\partial_1\widetilde u,\partial_2\widetilde u,0).$

### MSC:

 35Q30 Navier-Stokes equations 35B65 Smoothness and regularity of solutions to PDEs 35D30 Weak solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids

### Keywords:

Navier-Stokes equations; Beale-Kato-Majda criterion
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### References:

 [1] Leray, J., Essai sur les le mouvement d’un liquide visqueux emplissant l’espace, Acta. Math., 63, 193-248 (1934) [2] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-831 (1982) · Zbl 0509.35067 [3] Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational. Mech. Anal., 9, 187-195 (1962) · Zbl 0106.18302 [4] Struwe, M., On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41, 437-458 (1988) · Zbl 0632.76034 [5] Beirão da Veiga, H., A new regularity class for the Navier-Stokes equations in $$R^n$$, Chinese Ann. Math., 16, 407-412 (1995) · Zbl 0837.35111 [6] Beale, J.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94, 61-66 (1984) · Zbl 0573.76029 [7] Majda, A.; Bertozzi, A., Vorticity and Incompressible Flow (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0983.76001 [8] Kozono, H.; Taniuchi, Y., Limitting case of the Sobolev inequality in BMO with application to the Euler equations, Comm. Math. Phys., 214, 191-200 (2000) · Zbl 0985.46015 [9] 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 [10] Chen, Z.-M.; Xin, Z., Homogeneity criterion for the Navier-Stokes equations in the whole spaces, J. Math. Fluid Mech., 3, 152-182 (2001) · Zbl 0982.35081 [11] Kozono, H.; Yatsu, N., Extension criterion via two-components of vorticity on strong solution to the 3D Navier-Stokes equations, Math. Z., 246, 55-68 (2003) · Zbl 1060.35105 [12] Zhang, Z.; Chen, Q., Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $$R^3$$, J. Differential Equations, 216, 470-481 (2005) · Zbl 1091.35064 [13] Dong, B.-Q.; Chen, Z.-M., Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components, J. Math. Anal. Appl., 338, 1-10 (2008) · Zbl 1132.35432 [14] Zhou, Y., A new regularity criterion for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl., 84, 1496-1514 (2005) · Zbl 1092.35081 [15] He, C., Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electron. J. Diff. Equ., 49, 1-13 (2000) [16] Neustupa, J.; Penel, P., Regularity of a suitable weak solution to the Navier-Stokes equations as a consequence of regularity of one velocity component, (Applied Nonlinear Analysis (1999), Kluwer/Plenum: Kluwer/Plenum New York), 391-402 · Zbl 0953.35113 [17] Kukavica, I.; Ziane, M., One component regularity for the Navier-Stokes equations, Nonlinearity, 19, 453-469 (2006) · Zbl 1149.35069 [18] Cao, C.; Titi, E., Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57, 2643-2661 (2008) · Zbl 1159.35053 [19] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Fluids (1969), Gorden Brech: Gorden Brech New York · Zbl 0184.52603 [20] Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis (1977), North-Holland: North-Holland Amsterdam, New York · Zbl 0383.35057 [21] Stein, E. M., Harmonic Analysis: Real-Variable Mathods, Orthogonality, and Oscillatory Integrals (1993), Princeton University Press: Princeton University Press New Jersey · Zbl 0821.42001 [22] Chemin, J.-Y., Perfect Incompressible Fluids (1998), Oxford University Press: Oxford University Press New York · Zbl 0927.76002 [23] Fujita, H.; Kato, T., On the nonstationary Navier-Stokes initial value problem, Arch. Rational. Mech. Anal., 16, 269-315 (1964) · Zbl 0126.42301 [24] Serrin, J., The initial value problem for the Navier-Stokes equations, (Langer, R. E., Nonlinear Problems (1963), University of Wisconsin Press: University of Wisconsin Press Madison), 69-98 · Zbl 0115.08502 [25] Triebel, H., Theory of Function Spaces (1983), Birkhäuser Verlag: Birkhäuser Verlag Basel-Boston · Zbl 0546.46028
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