On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbb {R}^3$
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Abstract:
In this paper we establish a Serrin-type regularity criterion on the gradient of pressure for the weak solutions to the Navier-Stokes equations in $\mathbb {R}^3$. It is proved that if the gradient of pressure belongs to $L^{\alpha ,\gamma }$ with $2/\alpha +3/\gamma \leq 3$, $1\leq \gamma \leq \infty$, then the weak solution is actually regular. Moreover, we give a much simpler proof of the regularity criterion on the pressure, which was showed recently by Berselli and Galdi (Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585–3595).References
- Hugo Beirão da Veiga, On the smoothness of a class of weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech. 2 (2000), no. 4, 315–323. MR 1814220, DOI 10.1007/PL00000955
- Luigi C. Berselli and Giovanni P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585–3595. MR 1920038, DOI 10.1090/S0002-9939-02-06697-2
- L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. MR 673830, DOI 10.1002/cpa.3160350604
- Dongho Chae and Jihoon Lee, Regularity criterion in terms of pressure for the Navier-Stokes equations, Nonlinear Anal. 46 (2001), no. 5, Ser. A: Theory Methods, 727–735. MR 1857154, DOI 10.1016/S0362-546X(00)00163-2
- Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259, DOI 10.7208/chicago/9780226764320.001.0001
- Yoshikazu Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations 62 (1986), no. 2, 186–212. MR 833416, DOI 10.1016/0022-0396(86)90096-3
- Eberhard Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213–231 (German). MR 50423, DOI 10.1002/mana.3210040121
- L. Iskauriaza, G. A. Serëgin, and V. Shverak, $L_{3,\infty }$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3–44 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 2, 211–250. MR 1992563, DOI 10.1070/RM2003v058n02ABEH000609
- Tosio Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $\textbf {R}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR 760047, DOI 10.1007/BF01174182
- Hideo Kozono and Hermann Sohr, Regularity criterion of weak solutions to the Navier-Stokes equations, Adv. Differential Equations 2 (1997), no. 4, 535–554. MR 1441855
- Hideo Kozono and Yasushi Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z. 235 (2000), no. 1, 173–194. MR 1785078, DOI 10.1007/s002090000130
- J. Leray, Étude de divers équations intégrales nonlinearies et de quelques problemes que posent lhydrodinamique, J. Math. Pures. Appl., 12(1931), 1-82.
- Kyūya Masuda, Weak solutions of Navier-Stokes equations, Tohoku Math. J. (2) 36 (1984), no. 4, 623–646. MR 767409, DOI 10.2748/tmj/1178228767
- Mike O’Leary, Pressure conditions for the local regularity of solutions of the Navier-Stokes equations, Electron. J. Differential Equations (1998), Paper No. 12, 9. MR 1625358
- Vladimir Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math. 66 (1976), no. 2, 535–552. MR 454426, DOI 10.2140/pjm.1976.66.535
- James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187–195. MR 136885, DOI 10.1007/BF00253344
- Hermann Sohr, Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes, Math. Z. 184 (1983), no. 3, 359–375 (German). MR 716283, DOI 10.1007/BF01163510
- Michael Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 4, 437–458. MR 933230, DOI 10.1002/cpa.3160410404
- Roger Temam, Navier-Stokes equations, AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis; Reprint of the 1984 edition. MR 1846644, DOI 10.1090/chel/343
- Gang Tian and Zhouping Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom. 7 (1999), no. 2, 221–257. MR 1685610, DOI 10.4310/CAG.1999.v7.n2.a1
- Wolf von Wahl, Regularity of weak solutions of the Navier-Stokes equations, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 497–503. MR 843635
- Yong Zhou, Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain, Math. Ann. 328 (2004), no. 1-2, 173–192. MR 2030374, DOI 10.1007/s00208-003-0478-x
Additional Information
- Yong Zhou
- Affiliation: Department of Mathematics, East China Normal University, Shanghai, 200062, People’s Republic of China
- Email: yzhou@math.ecnu.edu.cn
- Received by editor(s): February 3, 2004
- Published electronically: August 19, 2005
- Communicated by: David S. Tartakoff
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 149-156
- MSC (2000): Primary 35B45, 35B65, 76D05
- DOI: https://doi.org/10.1090/S0002-9939-05-08312-7
- MathSciNet review: 2170554