American Institute of Mathematical Sciences

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August  2016, 36(8): 4477-4493. doi: 10.3934/dcds.2016.36.4477

On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity

Xiangdi Huang 1, and  Zhouping Xin 2,

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

Received  June 2015 Revised  October 2015 Published  March 2016

It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularity in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomena, it remains to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also gives an affirmative answer to a strong version of a problem proposed by J.Nash in 1950s.
Keywords: vacuum., Compressible Navier-Stokes system, continuation principle.
Mathematics Subject Classification: Primary: 35B65, 35Q35, 76N1.
Citation: Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477
References:
[1]

Y. Cho and B. J. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math.Anal. Appl., 320 (2006), 819-826. doi: 10.1016/j.jmaa.2005.08.005.  Google Scholar

[2]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differ. Eqns., 228 (2006), 377-411. doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[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), 337-350. doi: 10.1016/j.anihpc.2009.09.012.  Google Scholar

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D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354. doi: 10.1007/s002050050055.  Google Scholar

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D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, J.Differ Eqns., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

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X. D. Huang, Some Results on Blowup of Solutions to the Compressible Navier-Stokes Equations, Ph.D thesis, The Chinese University of Hong Kong, 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), 479-490. doi: 10.4310/MAA.2009.v16.n4.a4.  Google Scholar

[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), 147-171. doi: 10.1007/s00220-013-1791-1.  Google Scholar

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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), 1872-1886. doi: 10.1137/100814639.  Google Scholar

[13]

X. D. Huang, J. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35. doi: 10.1007/s00220-010-1148-y.  Google Scholar

[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), 549-585. doi: 10.1002/cpa.21382.  Google Scholar

[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), 671-686. doi: 10.1007/s11425-010-0042-6.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics.Compressible Models, New York, Oxford University Press, 2, 1998.  Google Scholar

[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), 67-104.  Google Scholar

[18]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France., 90 (1962), 487-497.  Google Scholar

[19]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer.J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar

[20]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa., 13 (1959), 115-162.  Google Scholar

[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), 1762-1774. doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[22]

D. Serre, Variations de grande amplitude pour la densite d'un fluide visqueux compressible, Phys. D., 48 (1991), 113-128. doi: 10.1016/0167-2789(91)90055-E.  Google Scholar

[23]

J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational Mech. Anal., 3 (1959), 271-288.  Google Scholar

[24]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  Google Scholar

[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), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[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), 727-742. doi: 10.1007/s00205-011-0407-1.  Google Scholar

[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), 1283-1316. doi: 10.1007/BF02106835.  Google Scholar

[28]

Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

[29]

Z. P. Xin and W. Yan, On Blowup of classical solutions to the compressible Navier-Stokes equations, Comm. Math. Phys., 321 (1995), 529-541. doi: 10.1007/s00220-012-1610-0.  Google Scholar

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), 819-826. doi: 10.1016/j.jmaa.2005.08.005.  Google Scholar

[2]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differ. Eqns., 228 (2006), 377-411. doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[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), 337-350. doi: 10.1016/j.anihpc.2009.09.012.  Google Scholar

[4]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.  Google Scholar

[5]

D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354. doi: 10.1007/s002050050055.  Google Scholar

[6]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, J.Differ Eqns., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[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, The Chinese University of Hong Kong, 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), 479-490. doi: 10.4310/MAA.2009.v16.n4.a4.  Google Scholar

[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), 147-171. doi: 10.1007/s00220-013-1791-1.  Google Scholar

[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), 1872-1886. doi: 10.1137/100814639.  Google Scholar

[13]

X. D. Huang, J. Li and Z. P. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35. doi: 10.1007/s00220-010-1148-y.  Google Scholar

[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), 549-585. doi: 10.1002/cpa.21382.  Google Scholar

[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), 671-686. doi: 10.1007/s11425-010-0042-6.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics.Compressible Models, New York, Oxford University Press, 2, 1998.  Google Scholar

[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), 67-104.  Google Scholar

[18]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France., 90 (1962), 487-497.  Google Scholar

[19]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer.J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar

[20]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa., 13 (1959), 115-162.  Google Scholar

[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), 1762-1774. doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[22]

D. Serre, Variations de grande amplitude pour la densite d'un fluide visqueux compressible, Phys. D., 48 (1991), 113-128. doi: 10.1016/0167-2789(91)90055-E.  Google Scholar

[23]

J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational Mech. Anal., 3 (1959), 271-288.  Google Scholar

[24]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  Google Scholar

[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), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[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), 727-742. doi: 10.1007/s00205-011-0407-1.  Google Scholar

[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), 1283-1316. doi: 10.1007/BF02106835.  Google Scholar

[28]

Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

[29]

Z. P. Xin and W. Yan, On Blowup of classical solutions to the compressible Navier-Stokes equations, Comm. Math. Phys., 321 (1995), 529-541. doi: 10.1007/s00220-012-1610-0.  Google Scholar

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