Electronic Journal of Differential Equations, Vol. 00(00), No. 05, pp. 5. ISSN: 07-669. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu REGUARITY OF GENERAIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VEOCITY YUWEN UO Abstract. In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms ( ) α u. It is proved that if div(u/ u ) p (0, T ; q (R 3 )) with p + 3 q 3, 6 α 3 < q. then any smooth on GNS in [0, T ) remains smooth on [0, T ].. Introduction We consider the incompressible generalized Navier-Stokes equation (GNS) u t + u u + ( ) α u = p u = 0 (.) Where u = u(x, t) denotes the velocity field, p = p(x, t) the scalar pressure and u 0 (x) with u 0 = 0 in the sense of distribution is the initial velocity field. The fractional power of aplace operator ( ) α is defined as in [3]: ( ) α f(ξ) = ξ ˆf(ξ). For notational convenience, we write Λ = ( ) /. When α =, (.) become the usual Navier-Stokes equations. Up to now, it is still unknown whether or not there exist global solution for Navier-Stokes equations even if the initial data is sufficiently smooth. This famous problem lead to extensively study the regularity of Navier-Stokes equations. There are plenty of literatures for usual Navier-Stokes equations, we mentioned some of them. For well-posedness, the readers could refer to eray[9], Kato[7], Cannone[3], Giga and Miyakawa[6] and Taylor[]. For regularity results, one could refer to Serrin[], Kozono and Sohr[8], Beale, Kato and Majda[], Constantin and Feffernan[5]. For general α, Wu[7] proved that if u 0, then the GNS (.) posses a weak solution u satisfying u ([0, T ]; ) ([0, T ]; H α ). Moreover, he showed that all solutions are global if α / + n/, where n is space dimension. For α < /+n/, Wu[8] studied the local well-posedness of (.) in Besov spaces. For 000 Mathematics Subject Classification. 35D0, 35Q35, 76D03. Key words and phrases. Generalized Navier-Stokes equation; regularity; Serrin criteria. c 00 Texas State University - San Marcos. Submitted April 8, 00. Published August, 00.
Y. UO EJDE-00/05 the regularity of GMHD equations, Wu[9] obtained some Serrin s type criterion. atter, Zhou[], Wu[6] and uo[0] improved some results of Wu. These results can also be used for GNS equations for GMHD equations contains GNS equations. In this short paper, we studied the regularity to GNS equations in terms of the direction of velocity which is used firstly by Vasseur[5]. He showed that if the initial value u 0 (R 3 ), and div(u/ u ) p (0, ; q (R 3 )) with p + 3 q, q 6, p then u is smooth on (0, ) R 3. atter, uo[] extended this result to MHD equations. The main result of this paper is as follows. Theorem.. et 3 < α < 3, u 0 H (R 3 ), u is a smooth solution of () in [0, T ). If div(u/ u ) p (0, T ; q (R 3 )) with then u remains smooth in [0, T ]. p + 3 q 3, 6 α 3 < q. To prove this theorem, we need the following result. emma.. With 0 < α <, θ, Λ α θ p with p = k we obtain θ p θλ α θdx Λ α p θ dx. p The proof is similar with Córdoba and Córdoba [], readers can find the details in Wu[0].. Proof of the main result Multiplying both side of the equations by u u, and integrating by parts, we obtain d dt u + u u ( ) α udx = p u u u dx (.) R 3 R 3 By emma., the left side satisfies d dt u + u u ( ) α udx d R dt u + Λ α u dx. (.) 3 R 3 So we obtain d dt u + Λα u dx p u u u dx. (.3) R 3 Taking the divergence of (.), one has p = i,j i j (u i, u j ), by Calderon-Zygmund inequality, we have p p C u p.
EJDE-00/05 REGUARITY OF NAVEIR-STOKES EQUATIONS 3 Then, using Hölder inequality, one obtains p u u u dx = p u u R 3 R u u dx 3 where /r + /q =. Here we used the fact p r u r u div(u/ u ) q C u r u div(u/ u ) q, u div(u/ u ) = u u u. By the interpolation inequality and the Sobolev embedding theorem [], we have where So we obtain R 3 p u u u u r C u θ u θ s = C u θ u θ/ s C u θ Λ α u θ/, θ + θ s = r, s = 6 3, (.) < r < s, 0 < θ <. (.5) u dx C u ( θ) Λ α u θ u div(u/ u ) q C u div(u/ u ) θ q u + Λα u, (.6) where the last inequality is deduced from Young s inequality. Combining (.)-(.6), we have d dt u + Λα u θ dx C u div(u/ u ) q u + Λα u. If u div(u/ u ) p,q and /( θ) p, then by Gronwell s inequality, we can claim that the smooth solution in [0, T ) remains smooth in [0, T ]. Now we search for the conditions which ensure u div(u/ u ) p,q and /( θ) p. Since θ (0, ) and r, q, s, θ satisfy we obtain That is, if then /( θ) p. r + q =, θ < r < s, s = + θ s = r, 6 3, θ = q q 3. p + 3 q,
Y. UO EJDE-00/05 et div(u/ u ) p,q. We know u ([0, T ]; ) ([0, T ]; H α ) and thus u a,b with /a + 3/b = 3/. So we obtain u div(u/ u ) p,q with p = a + p, q = b + q. From this relation, we obtain, if /p + 3/q 3/, then p + 3 q. That is, if p + 3 q 3, then u div(u/ u ) p,q 6. And the condition α 3 < q ensures the inequality q > 3, which implies < r < s, 0 < θ <, 6 s = 3. This completes the proof. References [] J. Beale, T. Kato and A. Majda; Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 9 (98), 6-66. [] J. Bergh and J. öfström; Interpolation Spaces, An Introduction. Springer-Verlag, Berlin, 976:5-53. [3] M. Cannone; A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 3 (997), 673-697. [] A. Códorba, D. Códorba; A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 9 (00), 5-58. [5] P. Constantin and Ch. Fefferman; Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J. (99), 775-789. [6] Y. Giga, T. Miyakawa; Navier-Stokes flow in R 3 with measures as initial vorticity and Morry spaces, Comm. PDE, (989), 577-68. [7] T. Kato; Strong solutions of the Navier-Stokes in R n with applications to weak solutions, Math. Zeit., 87 (98), 7-80. [8] H. Kozono and H. Sohr; Regularity criterion on weak solutions to the Navier-Stokes equations, Adv. Diff. Eqns No., 535-55. [9] J. eray; Sur le mouvement d um liquide visqieux emlissant l space, Acta Math. 63 (93), 93-8. [0] Y. uo; On the regularity of generalized MHD equations. J. Math. Anal. Appl., 365 (00), 806-808. [] Y. uo; Regularity of weak solutions to the magneto-hydrodynamics equations in terms of the direction of velocity. Electron. J. Diff. Equ., Vol. 009 (009), No. 3, pp. -7. [] J. Serrin; On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9 (96), 87-9. [3] E. M. Stein; Singular integrals and differentiability properties of functions [M]. Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 970: 6-. [] M. Taylor; Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. PDE 7 (99), 07-56. [5] A. Vasseur; Regularity criteria for 3D Navier-Stokes equations in terms of the direction of the velocity. Appl. Math., 5(009), No., 7-5. [6] G. Wu; Regularity criteria for the 3D generalized MHD equations in terms of vorticity. Nonlinear Analysis, TMA, 7 (009), 5-58. [7] J. Wu; Generalized MHD equations. J. Diff. Equations, 95 (003) 8-3. [8] J. Wu; The generalized incompressible Navier-Stokes equations in Besov spaces, Dyn. Partial Differ. Eq., (00), 38-00.
EJDE-00/05 REGUARITY OF NAVEIR-STOKES EQUATIONS 5 [9] J. Wu; Regularity Criteria for the Generalized MHD Equations, Communications in Partial Dierential Equations, 33 (008), 85-306. [0] J. Wu; ower bounds for an integral involving fractional aplacians and the generalized Navier-Stokes equations in Besov spaces. Commun. Math. Phys. 63(006), 803-83. [] Y. Zhou; Regularity criteria for the generalized viscous MHD equations. Ann. I. H. Poincaré, (007), 9-505. Yuwen uo School of Mathematics & Statistics, Chongqing University of Technology, Chongqing 00050, China E-mail address: petitevin@gmail.com