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

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.
Citation: Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems - A, 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.  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.  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.  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.  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.  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, (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.  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.  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.  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.  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.  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.  doi: 10.1007/s11425-010-0042-6.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics.Compressible Models,, New York, 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.   Google Scholar

[18]

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

[19]

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

[20]

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

[24]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.  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.  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.  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.  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.  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.  doi: 10.1007/s11425-010-0042-6.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics.Compressible Models,, New York, 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.   Google Scholar

[18]

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

[19]

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

[20]

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

[24]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.   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.  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.  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.  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.  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.  doi: 10.1007/s00220-012-1610-0.  Google Scholar

[1]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[2]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[3]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[4]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[5]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[6]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141

[7]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[8]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408

[9]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[10]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128

[11]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[12]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[13]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[14]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039

[15]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[16]

Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062

[17]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002

[18]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[19]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280

[20]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (115)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]