A blow-up criterion for compressible viscous heat-conductive flows
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 337-350.

Nous étudions un problème de valeur limite initiale pour les équations de Navier–Stokes tridimensionnelles des fluides visqueux conducteurs de chaleur dans un domaine délimité lisse. Nous établissons un critère d'explosion pour les solutions fortes en termes de température et de gradient de vitesse seulement, semblable au critère de Beale–Kato–Majda pour les écoulements incompressibles idéaux.

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.

DOI : 10.1016/j.anihpc.2009.09.012
Classification : 76N10, 35M10, 35Q30
Mots-clés : Blow-up criterion, Strong solutions, Compressible Navier–Stokes equations, Heat-conductive flows 
@article{AIHPC_2010__27_1_337_0,
     author = {Fan, Jishan and Jiang, Song and Ou, Yaobin},
     title = {A blow-up criterion for compressible viscous heat-conductive flows},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {337--350},
     publisher = {Elsevier},
     volume = {27},
     number = {1},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.09.012},
     mrnumber = {2580513},
     zbl = {1352.35109},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.012/}
}
TY  - JOUR
AU  - Fan, Jishan
AU  - Jiang, Song
AU  - Ou, Yaobin
TI  - A blow-up criterion for compressible viscous heat-conductive flows
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 337
EP  - 350
VL  - 27
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.012/
DO  - 10.1016/j.anihpc.2009.09.012
LA  - en
ID  - AIHPC_2010__27_1_337_0
ER  - 
%0 Journal Article
%A Fan, Jishan
%A Jiang, Song
%A Ou, Yaobin
%T A blow-up criterion for compressible viscous heat-conductive flows
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 337-350
%V 27
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.012/
%R 10.1016/j.anihpc.2009.09.012
%G en
%F AIHPC_2010__27_1_337_0
Fan, Jishan; Jiang, Song; Ou, Yaobin. A blow-up criterion for compressible viscous heat-conductive flows. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 337-350. doi : 10.1016/j.anihpc.2009.09.012. http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.012/

[1] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984), 61-66 | MR | Zbl

[2] Y. Cho, H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations 228 (2006), 377-411 | MR | Zbl

[3] J. Fan, S. Jiang, Blow-up criteria for the Navier–Stokes equations of compressible fluids, J. Hyperbolic Differ. Equ. 5 (2008), 167-185 | MR | Zbl

[4] E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier–Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech. 3 (2001), 358-392 | MR | Zbl

[5] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Univ. Press, Oxford (2004) | MR | Zbl

[6] E. Feireisl, On the motion of a viscous, compressible and heat conducting fluid, Indiana Univ. Math. J. 53 (2004), 1705-1738 | MR | Zbl

[7] D. Hoff, Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Ration. Mech. Anal. 139 (1997), 303-354 | MR | Zbl

[8] D. Hoff, Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech. 7 (2005), 315-338 | MR | Zbl

[9] X. Huang, Z. Xin, A blow-up criterion for classical solutions to the compressible Navier–Stokes equations, 19 March, 2009, arXiv:0903.3090v2 [math-ph] | MR

[10] S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm. Math. Phys. 178 (1996), 339-374 | MR | Zbl

[11] S. Jiang, P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier–Stokes equations, Comm. Math. Phys. 215 (2001), 559-581 | MR | Zbl

[12] S. Jiang, P. Zhang, Axisymmetric solutions of the 3-D Navier–Stokes equations for compressible isentropic fluids, J. Math. Pures Appl. 82 (2003), 949-973 | MR | Zbl

[13] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2, Oxford Lecture Ser. Math. Appl. vol. 10, Clarendon Press, Oxford (1998) | MR | Zbl

[14] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), 67-104 | MR | Zbl

[15] A. Matsumura, T. Nishida, The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids, Comm. Math. Phys. 89 (1983), 445-464 | MR | Zbl

[16] O. Rozanova, Blow up of smooth solutions to the compressible Navier–Stokes equations with the data highly decreasing at infinity, J. Differential Equations 245 (2008), 1762-1774 | MR | Zbl

[17] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth, Heidelberg (1995) | MR | Zbl

[18] Z. Xin, Blow up of smooth solutions to the compressible Navier–Stokes equation with compact density, Comm. Pure Appl. Math. 51 (1998), 229-240 | MR | Zbl

Cité par Sources :