| Title:
|
A geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations (English) |
| Author:
|
Neustupa, Jiří |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
685-698 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We deal with a suitable weak solution $(\bold v,p)$ to the Navier-Stokes equations in a domain $\Omega \subset \mathbb R^3$. We refine the criterion for the local regularity of this solution at the point $(\bold fx_0,t_0)$, which uses the $L^3$-norm of $\bold v$ and the $L^{3/2}$-norm of $p$ in a shrinking backward parabolic neighbourhood of $(\bold x_0,t_0)$. The refinement consists in the fact that only the values of $\bold v$, respectively $p$, in the exterior of a space-time paraboloid with vertex at $(\bold x_0,t_0)$, respectively in a ”small” subset of this exterior, are considered. The consequence is that a singularity cannot appear at the point $(\bold x_0,t_0)$ if $\bold v$ and $p$ are “smooth” outside the paraboloid. (English) |
| Keyword:
|
Navier-Stokes equation |
| Keyword:
|
suitable weak solution |
| Keyword:
|
regularity |
| MSC:
|
35B65 |
| MSC:
|
35Q30 |
| MSC:
|
76D03 |
| MSC:
|
76D05 |
| idZBL:
|
Zbl 06433692 |
| idMR:
|
MR3306858 |
| DOI:
|
10.21136/MB.2014.144145 |
| . |
| Date available:
|
2015-02-04T09:31:12Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144145 |
| . |
| Reference:
|
[1] Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations.Commun. Pure Appl. Math. 35 (1982), 771-831. Zbl 0509.35067, MR 0673830, 10.1002/cpa.3160350604 |
| Reference:
|
[2] Farwig, R., Kozono, H., Sohr, H.: Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition.Parabolic and Navier-Stokes Equations. Part 1. Proceedings of the confererence, Będlewo, Poland, 2006 Banach Center Publ. 81 Polish Academy of Sciences, Institute of Mathematics, Warsaw (2008), 175-184 J. Rencławowicz et al. Zbl 1154.35416, MR 2549330 |
| Reference:
|
[3] Farwig, R., Kozono, H., Sohr, H.: An $L^q$-approach to Stokes and Navier-Stokes equations in general domains.Acta Math. 195 (2005), 21-53. Zbl 1111.35033, MR 2233684, 10.1007/BF02588049 |
| Reference:
|
[4] Kučera, P., Skalák, Z.: A note on the generalized energy inequality in the Navier-Stokes equations.Appl. Math., Praha 48 (2003), 537-545. Zbl 1099.35099, MR 2025962, 10.1023/B:APOM.0000024492.23444.29 |
| Reference:
|
[5] Ladyzhenskaya, O. A., Seregin, G. A.: On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations.J. Math. Fluid Mech. 1 (1999), 356-387. Zbl 0954.35129, MR 1738171, 10.1007/s000210050015 |
| Reference:
|
[6] Lin, F.: A new proof of the Caffarelli-Kohn-Nirenberg theorem.Commun. Pure Appl. Math. 51 (1998), 241-257. Zbl 0958.35102, MR 1488514, 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A |
| Reference:
|
[7] Nečas, J., Neustupa, J.: New conditions for local regularity of a suitable weak solution to the Navier-Stokes equation.J. Math. Fluid Mech. 4 (2002), 237-256. Zbl 1010.35081, MR 1932862, 10.1007/s00021-002-8544-9 |
| Reference:
|
[8] Neustupa, J.: A note on local interior regularity of a suitable weak solution to the Navier-Stokes problem.Discrete Contin. Dyn. Syst., Ser. S 6 (2013), 1391-1400. Zbl 1260.35125, MR 3039705, 10.3934/dcdss.2013.6.1391 |
| Reference:
|
[9] Neustupa, J.: A refinement of the local Serrin-type regularity criterion for a suitable weak solution to the Navier-Stokes equations.Arch. Ration. Mech. Anal. 214 (2014), 525-544. Zbl 1304.35502, MR 3255699, 10.1007/s00205-014-0761-x |
| Reference:
|
[10] Neustupa, J.: A removable singularity in a suitable weak solution to the Navier-Stokes equations.Nonlinearity 25 (2012), 1695-1708. Zbl 1245.35085, MR 2924731, 10.1088/0951-7715/25/6/1695 |
| Reference:
|
[11] Scheffer, V.: Hausdorff measure and the Navier-Stokes equations.Commun. Math. Phys. 55 (1977), 97-112. Zbl 0357.35071, MR 0510154, 10.1007/BF01626512 |
| Reference:
|
[12] Seregin, G., Šverák, V.: On smoothness of suitable weak solutions to the Navier-Stokes equations.J. Math. Sci., New York 130 (2005), 4884-4892 translated from Zap. Nauchn. Semin. POMI 306 (2003), 186-198. MR 2065503, 10.1007/s10958-005-0383-9 |
| Reference:
|
[13] Seregin, G. A.: Local regularity for suitable weak solutions of the Navier-Stokes equations.Russ. Math. Surv. 62 (2007), 595-614 translated from Usp. Mat. Nauk 62 149-168 (2007). Zbl 1139.76018, MR 2355422, 10.1070/RM2007v062n03ABEH004415 |
| Reference:
|
[14] Sohr, H., Wahl, W. von: On the regularity of the pressure of weak solutions of Navier-Stokes equations.Arch. Math. 46 (1986), 428-439. MR 0847086, 10.1007/BF01210782 |
| Reference:
|
[15] Talenti, G.: Best constant in Sobolev inequality.Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. Zbl 0353.46018, MR 0463908 |
| Reference:
|
[16] Wolf, J.: A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations.Advances in Mathematical Fluid Mechanics. Selected papers of the international conference on mathematical fluid mechanics, Estoril, Portugal, 2007 Springer Berlin 613-630 (2010), R. Rannacher et al. MR 2665054 |
| . |
