| Title:
|
Interior regularity of weak solutions to the perturbed Navier-Stokes equations (English) |
| Author:
|
Han, Pigong |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
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0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
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57 |
| Issue:
|
5 |
| Year:
|
2012 |
| Pages:
|
427-444 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69 (1990), 237–254. (English) |
| Keyword:
|
perturbed Navier-Stokes equations |
| Keyword:
|
interior regularity |
| Keyword:
|
partial regularity |
| MSC:
|
35B65 |
| MSC:
|
35Q30 |
| MSC:
|
76D05 |
| idZBL:
|
Zbl 1265.35246 |
| idMR:
|
MR2984612 |
| DOI:
|
10.1007/s10492-012-0025-8 |
| . |
| Date available:
|
2012-08-19T21:54:28Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142908 |
| . |
| Reference:
|
[1] Batchelor, G. K.: An Introduction to Fluid Dynamics.Cambridge University Press Cambridge (1967). Zbl 0152.44402, MR 1744638 |
| Reference:
|
[2] Bae, H.-O., Choe, H. J.: A regularity criterion for the Navier-Stokes equations.Commun. Partial. Differ. Equations 32 (2007), 1173-1187. Zbl 1220.35111, MR 2354489, 10.1080/03605300701257500 |
| Reference:
|
[3] Veiga, H. Beirao da: On the smoothness of a class of weak solutions to the Navier-Stokes equations.J. Math. Fluid Mech. 2 (2000), 315-323. MR 1814220, 10.1007/PL00000955 |
| Reference:
|
[4] Veiga, H. Beirao da: A new regularity class for the Navier-Stokes equations in $\Bbb R^n$.Chin. Ann. Math., Ser. B 16 (1995), 407-412. MR 1380578 |
| Reference:
|
[5] Veiga, H. Beirao da: A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations.J. Math. Fluid Mech. 2 (2000), 99-106. MR 1765772, 10.1007/PL00000949 |
| Reference:
|
[6] Berselli, L. C., Galdi, G. P.: Regularity criteria involving the pressure for the weak solutions of the Navier-Stokes equations.Proc. Am. Math. Soc. 130 (2002), 3585-3595. MR 1920038, 10.1090/S0002-9939-02-06697-2 |
| Reference:
|
[7] 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:
|
[8] Cao, C., Titi, E.: Regularity criteria for the three-dimensional Navier-Stokes equations.Indiana Univ. Math. J. 57 (2008), 2643-2661. Zbl 1159.35053, MR 2482994, 10.1512/iumj.2008.57.3719 |
| Reference:
|
[9] Chae, D., Choe, H.-J.: Regularity of solutions to the Navier-Stokes equation.Electron. J. Differ. Equ. 5 (1999), 1-7. Zbl 0923.35117, MR 1673067 |
| Reference:
|
[10] Chae, D., Lee, J.: Regularity criterion in terms of pressure for the Navier-Stokes equations.Nonlinear Anal., Theory Methods Appl. 46 (2001), 727-735. Zbl 1007.35064, MR 1857154, 10.1016/S0362-546X(00)00163-2 |
| Reference:
|
[11] Chester, W.: A general theory for the motion of a body through a fluid at low Reynolds number.Proc. R. Soc. Lond., Ser. A 430 (1990), 89-104. Zbl 0703.76026, MR 1068486 |
| Reference:
|
[12] Chen, Z., Miyakawa, T.: Decay properties of weak solutions to a perturbed Navier-Stokes system in $\Bbb{R}^n$.Adv. Math. Sci. Appl. 7 (1997), 741-770. MR 1476275 |
| Reference:
|
[13] Farwig, R., Komo, C.: Regularity of weak solutions to the Navier-Stokes equations in exterior domains.NoDEA, Nonlinear Differ. Equ. Appl. 17 (2010), 303-321. Zbl 1189.76115, MR 2652230, 10.1007/s00030-010-0055-4 |
| Reference:
|
[14] Farwig, R., Kozono, H., Sohr, H.: Local in time regularity properties of the Navier-Stokes equations.Indiana Univ. Math. J. 56 (2007), 2111-2132. Zbl 1175.35100, MR 2359725, 10.1512/iumj.2007.56.3098 |
| Reference:
|
[15] Han, P.: Regularity of weak solutions to 3D incompressible Navier-Stokes equations.J. Evol. Equ. 10 (2010), 195-204. Zbl 1239.35110, MR 2602932, 10.1007/s00028-009-0045-3 |
| Reference:
|
[16] He, C.: Regularity for solutions to the Navier-Stokes equations with one velocity component regular.Electron J. Differ. Equ. 29 (2002), 1-13. Zbl 0993.35072, MR 1907705 |
| Reference:
|
[17] Hishida, T.: An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle.Arch. Ration. Mech. Anal. 150 (1999), 307-348. Zbl 0949.35106, MR 1741259, 10.1007/s002050050190 |
| Reference:
|
[18] Iskauriaza, L., Seregin, G., Shverak, V.: $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness.Uspekhi Mat. Nauk. 58 (2003), 3-44 Russian; translation in Russ. Math. Surv. 58 (2003), 211-250. MR 1992563 |
| Reference:
|
[19] Ladyzhenskaya, O., 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:
|
[20] 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:
|
[21] Masuda, K.: Weak solutions of the Navier-Stokes equations.Tôhoku Math. J., II. Ser. 36 (1984), 623-646. MR 0767409, 10.2748/tmj/1178228767 |
| Reference:
|
[22] Neustupa, J., Novotný, A., Penel, P.: An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity.Quaderni di Matematica, Vol. 10: Topics in Mathematical Fluid Mechanics G. P. Galdi, R. Rannacher (2003), 168-183 \MR 20517774. MR 2051774 |
| Reference:
|
[23] 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 Anal. A. Sequeira et al. Kluwer Academic/Plenum Publishers New York (1999), 391-402. Zbl 0953.35113, MR 1727461 |
| Reference:
|
[24] Penel, P., Pokorný, M.: Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity.Appl. Math. 49 (2004), 483-493. Zbl 1099.35101, MR 2086090, 10.1023/B:APOM.0000048124.64244.7e |
| Reference:
|
[25] Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations.Pac. J. Math. 66 (1976), 535-552. Zbl 0325.35064, MR 0454426, 10.2140/pjm.1976.66.535 |
| Reference:
|
[26] Seregin, G.: Navier-Stokes equations: almost $L_{3,\infty}$-case.J. Math. Fluid Mech. 9 (2007), 34-43. Zbl 1128.35085, MR 2305824, 10.1007/s00021-005-0190-6 |
| Reference:
|
[27] Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations.Arch. Ration. Mech. Anal. 9 (1962), 187-195. Zbl 0106.18302, MR 0136885, 10.1007/BF00253344 |
| Reference:
|
[28] Solonnikov, V. A.: Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations.Am. Math. Soc., Transl., II. Sér. 75 (1968), 1-116. Zbl 0187.03402 |
| Reference:
|
[29] Struwe, M.: On partial regularity results for the Navier-Stokes equations.Commun. Pure Appl. Math. 41 (1988), 437-458. Zbl 0632.76034, MR 0933230, 10.1002/cpa.3160410404 |
| Reference:
|
[30] Takahashi, S.: On interior regularity criteria for weak solutions of the Navier-Stokes equations.Manuscr. Math. 69 (1990), 237-254. Zbl 0718.35022, MR 1078355, 10.1007/BF02567922 |
| Reference:
|
[31] Zhou, Y.: Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain.Math. Ann. 328 (2004), 173-192. Zbl 1054.35062, MR 2030374, 10.1007/s00208-003-0478-x |
| Reference:
|
[32] Zhou, Y.: A new regularity criterion for weak solutions to the Navier-Stokes equations.J. Math. Pures Appl., IX. Sér. 84 (2005), 1496-1514. Zbl 1092.35081, MR 2181458, 10.1016/j.matpur.2005.07.003 |
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