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

Xiangdi  Huang; Zhouping Xin

doi:10.3934/dcds.2016.36.4477

Article Contents

2016, Volume 36, Issue 8: 4477-4493. Doi: 10.3934/dcds.2016.36.4477

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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
2.
The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Received: May 31, 2015

Revised: September 30, 2015

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B65, 35Q35, 76N10.

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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.
[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.
[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.
[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), 303-354.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), 215-254.doi: 10.1006/jdeq.1995.1111.
[7]	B. Haspot, Regularity of weak solutions of the compressible barotropic Navier-Stokes equations, preprint, arXiv:1001.1581
[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.
[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.
[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, arXiv:1107.4655.
[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.
[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.
[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.
[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.
[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.
[16]	P. L. Lions, Mathematical Topics in Fluid Mechanics.Compressible Models, New York, Oxford University Press, 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), 67-104.
[18]	J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France., 90 (1962), 487-497.
[19]	J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer.J. Math., 80 (1958), 931-954.doi: 10.2307/2372841.
[20]	L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa., 13 (1959), 115-162.
[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.
[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.
[23]	J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational Mech. Anal., 3 (1959), 271-288.
[24]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.
[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.
[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.
[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.
[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.
[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.