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On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity
1. | NCMIS, AMSS, Institute of Mathematics, Chinese Academy of Sciences, Beijing 100190, China |
2. | The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong |
References:
[1] |
Y. Cho and B. J. Jin, Blow-up of viscous heat-conducting compressible flows,, J. Math.Anal. Appl., 320 (2006), 819.
doi: 10.1016/j.jmaa.2005.08.005. |
[2] |
Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Differ. Eqns., 228 (2006), 377.
doi: 10.1016/j.jde.2006.05.001. |
[3] |
J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows,, Annales de l'Institut Henri Poincare (C) Analyse non lineaire., 27 (2010), 337.
doi: 10.1016/j.anihpc.2009.09.012. |
[4] |
E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).
|
[5] |
D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids,, Arch. Rational Mech. Anal., 139 (1997), 303.
doi: 10.1007/s002050050055. |
[6] |
D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data,, J.Differ Eqns., 120 (1995), 215.
doi: 10.1006/jdeq.1995.1111. |
[7] |
B. Haspot, Regularity of weak solutions of the compressible barotropic Navier-Stokes equations,, preprint, (). Google Scholar |
[8] |
X. D. Huang, Some Results on Blowup of Solutions to the Compressible Navier-Stokes Equations,, Ph.D thesis, (2009). Google Scholar |
[9] |
X. D. Huang and J. Li, On breakdown of solutions to the full compressible Navier-Stokes equations,, Methods Appl. Anal., 16 (2009), 479.
doi: 10.4310/MAA.2009.v16.n4.a4. |
[10] |
X. D. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations.,, preprint, (). Google Scholar |
[11] |
X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows,, Comm. Math. Phys., 324 (2013), 147.
doi: 10.1007/s00220-013-1791-1. |
[12] |
X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows,, Siam J. Math. Anal., 43 (2011), 1872.
doi: 10.1137/100814639. |
[13] |
X. D. Huang, J. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states,, Comm. Math. Phys., 301 (2011), 23.
doi: 10.1007/s00220-010-1148-y. |
[14] |
X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, Comm. Pure Appl. Math., 65 (2012), 549.
doi: 10.1002/cpa.21382. |
[15] |
X. D. Huang and Z. P. Xin, A blow-up criterion for classical solutions to the compressible Navier-Stokes equations,, Sci. in China., 53 (2010), 671.
doi: 10.1007/s11425-010-0042-6. |
[16] |
P. L. Lions, Mathematical Topics in Fluid Mechanics.Compressible Models,, New York, 2 (1998).
|
[17] |
A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.
|
[18] |
J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France., 90 (1962), 487.
|
[19] |
J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer.J. Math., 80 (1958), 931.
doi: 10.2307/2372841. |
[20] |
L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa., 13 (1959), 115.
|
[21] |
O. Rozanova, Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity,, J.Differ Eqns., 245 (2008), 1762.
doi: 10.1016/j.jde.2008.07.007. |
[22] |
D. Serre, Variations de grande amplitude pour la densite d'un fluide visqueux compressible,, Phys. D., 48 (1991), 113.
doi: 10.1016/0167-2789(91)90055-E. |
[23] |
J. Serrin, On the uniqueness of compressible fluid motion,, Arch. Rational Mech. Anal., 3 (1959), 271.
|
[24] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.
|
[25] |
Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations,, J. Math. Pures Appl., 95 (2011), 36.
doi: 10.1016/j.matpur.2010.08.001. |
[26] |
Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows,, Arch. Rational Mech. Anal., 201 (2011), 727.
doi: 10.1007/s00205-011-0407-1. |
[27] |
V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid,, Sib. Math. J., 36 (1995), 1283.
doi: 10.1007/BF02106835. |
[28] |
Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229.
doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. |
[29] |
Z. P. Xin and W. Yan, On Blowup of classical solutions to the compressible Navier-Stokes equations,, Comm. Math. Phys., 321 (1995), 529.
doi: 10.1007/s00220-012-1610-0. |
show all references
References:
[1] |
Y. Cho and B. J. Jin, Blow-up of viscous heat-conducting compressible flows,, J. Math.Anal. Appl., 320 (2006), 819.
doi: 10.1016/j.jmaa.2005.08.005. |
[2] |
Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Differ. Eqns., 228 (2006), 377.
doi: 10.1016/j.jde.2006.05.001. |
[3] |
J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows,, Annales de l'Institut Henri Poincare (C) Analyse non lineaire., 27 (2010), 337.
doi: 10.1016/j.anihpc.2009.09.012. |
[4] |
E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).
|
[5] |
D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids,, Arch. Rational Mech. Anal., 139 (1997), 303.
doi: 10.1007/s002050050055. |
[6] |
D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data,, J.Differ Eqns., 120 (1995), 215.
doi: 10.1006/jdeq.1995.1111. |
[7] |
B. Haspot, Regularity of weak solutions of the compressible barotropic Navier-Stokes equations,, preprint, (). Google Scholar |
[8] |
X. D. Huang, Some Results on Blowup of Solutions to the Compressible Navier-Stokes Equations,, Ph.D thesis, (2009). Google Scholar |
[9] |
X. D. Huang and J. Li, On breakdown of solutions to the full compressible Navier-Stokes equations,, Methods Appl. Anal., 16 (2009), 479.
doi: 10.4310/MAA.2009.v16.n4.a4. |
[10] |
X. D. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations.,, preprint, (). Google Scholar |
[11] |
X. D. Huang and J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows,, Comm. Math. Phys., 324 (2013), 147.
doi: 10.1007/s00220-013-1791-1. |
[12] |
X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows,, Siam J. Math. Anal., 43 (2011), 1872.
doi: 10.1137/100814639. |
[13] |
X. D. Huang, J. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states,, Comm. Math. Phys., 301 (2011), 23.
doi: 10.1007/s00220-010-1148-y. |
[14] |
X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, Comm. Pure Appl. Math., 65 (2012), 549.
doi: 10.1002/cpa.21382. |
[15] |
X. D. Huang and Z. P. Xin, A blow-up criterion for classical solutions to the compressible Navier-Stokes equations,, Sci. in China., 53 (2010), 671.
doi: 10.1007/s11425-010-0042-6. |
[16] |
P. L. Lions, Mathematical Topics in Fluid Mechanics.Compressible Models,, New York, 2 (1998).
|
[17] |
A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.
|
[18] |
J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France., 90 (1962), 487.
|
[19] |
J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer.J. Math., 80 (1958), 931.
doi: 10.2307/2372841. |
[20] |
L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa., 13 (1959), 115.
|
[21] |
O. Rozanova, Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity,, J.Differ Eqns., 245 (2008), 1762.
doi: 10.1016/j.jde.2008.07.007. |
[22] |
D. Serre, Variations de grande amplitude pour la densite d'un fluide visqueux compressible,, Phys. D., 48 (1991), 113.
doi: 10.1016/0167-2789(91)90055-E. |
[23] |
J. Serrin, On the uniqueness of compressible fluid motion,, Arch. Rational Mech. Anal., 3 (1959), 271.
|
[24] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.
|
[25] |
Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations,, J. Math. Pures Appl., 95 (2011), 36.
doi: 10.1016/j.matpur.2010.08.001. |
[26] |
Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows,, Arch. Rational Mech. Anal., 201 (2011), 727.
doi: 10.1007/s00205-011-0407-1. |
[27] |
V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid,, Sib. Math. J., 36 (1995), 1283.
doi: 10.1007/BF02106835. |
[28] |
Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229.
doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C. |
[29] |
Z. P. Xin and W. Yan, On Blowup of classical solutions to the compressible Navier-Stokes equations,, Comm. Math. Phys., 321 (1995), 529.
doi: 10.1007/s00220-012-1610-0. |
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