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A note on regularity criterion for the 3D Boussinesq system with partial viscosity. (English) Zbl 1171.35342
Summary: We prove a regularity criterion for $\omega := \text{curl }u\in L^1(0,T;\dot B^0_{\infty,\infty}) $ for the 3D Boussinesq system with partial viscosity. Here $u$ is the velocity, $\omega $ is the vorticity and $\dot B^0_{\infty,\infty}$ denotes the homogeneous Besov space.

35B65Smoothness and regularity of solutions of PDE
35Q35PDEs in connection with fluid mechanics
Full Text: DOI
[1] Beale, J. T.; 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 · doi:10.1007/BF01212349
[2] Cannon, J. R.; Dibenedetto, E.: The initial problem for the Boussinesq equation with data in lp, Lecture note in mathematics 771, 129-144 (1980) · Zbl 0429.35059
[3] Chae, D.: On the well-posedness of the Euler equations in the Besov and Triebel-Lizorkin spaces, , 42-57 (2002)
[4] Chae, D.: Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. anal. 38, No. 3-4, 339-358 (2004) · Zbl 1068.35097
[5] Chae, D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. math. 203, 497-513 (2006) · Zbl 1100.35084 · doi:10.1016/j.aim.2005.05.001
[6] David, G.; Jouré, J. L.: A boundedness criterion for generalized Calderón--Zygmund operators, Ann. math. 120, 371-397 (1994) · Zbl 0567.47025 · doi:10.2307/2006946
[7] J. Fan, T. Ozawa, Regularity conditions for the 3D Boussinesq equations with partial viscosity terms (2007) (under review)
[8] Ishimura, N.; Morimoto, H.: Remarks on the blow up criterion for the 3D Boussinesq equations, M3as 9, 1323-1332 (1999) · Zbl 1034.35113 · doi:10.1142/S0218202599000580
[9] Jawerth, B.: Some observations on Besov and Lizorkin--Triebel spaces, Math. scand. 40, 94-104 (1977) · Zbl 0358.46023
[10] Kato, T.; Ponce, G.: Commutator estimates and the Euler and Navier--Stokes equations, Commun. pure appl. Math. 41, 891-907 (1988) · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[11] Kozono, H.; Ogawa, T.; Taniuchi, Y.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z. 242, 251-278 (2002) · Zbl 1055.35087 · doi:10.1007/s002090100332
[12] Majda, A.: Introduction to pdes and waves for the atmosphere and ocean, Courant lecture notes in mathematics 9 (2003) · Zbl 1278.76004
[13] Triebel, H.: Theory of functions spaces II, (1992) · Zbl 0763.46025