| Title:
|
On the blow up criterion for the 2-D compressible Navier-Stokes equations (English) |
| Author:
|
Jiang, Lingyu |
| Author:
|
Wang, Yidong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
195-209 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Motivated by [10], we prove that the upper bound of the density function $\rho $ controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain. (English) |
| Keyword:
|
compressible Navier-Stokes equations |
| Keyword:
|
classical solutions |
| Keyword:
|
blow up criterion |
| MSC:
|
35B44 |
| MSC:
|
35Q30 |
| MSC:
|
35Q35 |
| MSC:
|
76D03 |
| idZBL:
|
Zbl 1224.35317 |
| idMR:
|
MR2595083 |
| . |
| Date available:
|
2010-07-20T16:27:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140562 |
| . |
| Reference:
|
[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 (1984), 61-66. Zbl 0573.76029, MR 0763762, 10.1007/BF01212349 |
| Reference:
|
[2] Choe, H. J., Jin, B. J.: Regularity of weak solutions of the compressible Navier-Stokes equations.J. Korean Math. Soc. 40 (2003), 1031-1050. Zbl 1034.76049, MR 2013486, 10.4134/JKMS.2003.40.6.1031 |
| Reference:
|
[3] Desjardins, B.: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations.Comm. P. D. E. 22 (1997), 977-1008. Zbl 0885.35089, MR 1452175, 10.1080/03605309708821291 |
| Reference:
|
[4] Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations.J. Math. Fluid Mech. 3 (2001), 358-392. MR 1867887, 10.1007/PL00000976 |
| Reference:
|
[5] Itaya, N.: On the Cauchy problem for the system of fundamental equations describing movement of compressible viscous fluids.K$\bar o$dai Math. Sem. Rep. 23 (1971), 60-120. MR 0283426, 10.2996/kmj/1138846265 |
| Reference:
|
[6] Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations.Comm. Math. Phys. 214 (2000), 191-200. Zbl 0985.46015, MR 1794270, 10.1007/s002200000267 |
| Reference:
|
[7] Lions, P. L.: Mathematical Topics in Fluid Mechanics, Vol 2. Compressible Models.Oxford lecture series in Mathematics and its Applications, 10, Oxford Sciences Publications. The Clarendon Press, Oxford University Press, New York (1998). Zbl 0908.76004, MR 1637634 |
| Reference:
|
[8] Tani, A.: On the first initial-boundary value problem of compressible viscous fluid motion.Publ. RIMS. Kyoto Univ. 13 (1977), 193-253. Zbl 0366.35070, 10.2977/prims/1195190106 |
| Reference:
|
[9] Xin, Z. P.: Blow up of smooth solutions to the compressible Navier-Stokes equation with compact density.Comm. Pure Appl. Math. 51 (1998), 229-240. MR 1488513, 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C |
| Reference:
|
[10] Vaigant, V. A., Kazhikhov, A. V.: On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid.Russian Sibirsk. Mat. Zh. 36 (1995), 1283-1316 translation in it Siberian Math. J. {\it 36} (1995). MR 1375428 |
| . |
