American Institute of Mathematical Sciences

Advanced
February  2016, 36(2): 1085-1103. doi: 10.3934/dcds.2016.36.1085

On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

Peixin Zhang 1, , Jianwen Zhang 2, and  Junning Zhao 2,

1. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
2. 

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received  March 2014 Revised  November 2014 Published  August 2015

In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.
Keywords: global classical solution, vacuum., large initial data, Compressible Navier-Stokes equations.
Mathematics Subject Classification: Primary: 76N10; Secondary: 35M1.
Citation: Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
References:
[1]

R. A. Adams, Sobolev Space,, Academic Press, (1975).   Google Scholar

[2]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid,, J. Math. Pures Appl., 83 (2004), 243.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[3]

Y. Cho and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Differ. Eqns., 190 (2003), 504.  doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative intial densities,, Manuscript Math., 120 (2006), 91.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[5]

E. Feiresl, A. Novotny and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358.  doi: 10.1007/PL00000976.  Google Scholar

[6]

X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional compressible flows,, SIAM J. Math. Anal., 43 (2011), 1872.  doi: 10.1137/100814639.  Google Scholar

[7]

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

[8]

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

[9]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Arch. Rational Mech. Anal., 132 (1995), 1.  doi: 10.1007/BF00390346.  Google Scholar

[10]

J. Leray, Sur le mouvement d'un kiquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[11]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids,, $2^{nd}$ edition, (1969).   Google Scholar

[12]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, (French),, Gauthier-Villars, (1969).   Google Scholar

[13]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2 Compressible Models,, Oxford University Press, (1998).   Google Scholar

[14]

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

[15]

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

[16]

R. Salvi and I. Straskraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty$,, J. Fac. Sci. Univ. Tokyo Sect. IA. Math., 40 (1993), 17.   Google Scholar

[17]

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

[18]

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

[19]

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

[20]

A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations,, Diff. Equations, 36 (2000), 701.  doi: 10.1007/BF02754229.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Space,, Academic Press, (1975).   Google Scholar

[2]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid,, J. Math. Pures Appl., 83 (2004), 243.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[3]

Y. Cho and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Differ. Eqns., 190 (2003), 504.  doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative intial densities,, Manuscript Math., 120 (2006), 91.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[5]

E. Feiresl, A. Novotny and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358.  doi: 10.1007/PL00000976.  Google Scholar

[6]

X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional compressible flows,, SIAM J. Math. Anal., 43 (2011), 1872.  doi: 10.1137/100814639.  Google Scholar

[7]

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

[8]

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

[9]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Arch. Rational Mech. Anal., 132 (1995), 1.  doi: 10.1007/BF00390346.  Google Scholar

[10]

J. Leray, Sur le mouvement d'un kiquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[11]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids,, $2^{nd}$ edition, (1969).   Google Scholar

[12]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, (French),, Gauthier-Villars, (1969).   Google Scholar

[13]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2 Compressible Models,, Oxford University Press, (1998).   Google Scholar

[14]

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

[15]

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

[16]

R. Salvi and I. Straskraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty$,, J. Fac. Sci. Univ. Tokyo Sect. IA. Math., 40 (1993), 17.   Google Scholar

[17]

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

[18]

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

[19]

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

[20]

A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations,, Diff. Equations, 36 (2000), 701.  doi: 10.1007/BF02754229.  Google Scholar

[1]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
[2]

Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041
[3]

Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072
[4]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[5]

Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127
[6]

Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353
[7]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122
[8]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521
[9]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[10]

Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure & Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987
[11]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
[12]

Ben Duan, Zhen Luo. Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2543-2564. doi: 10.3934/cpaa.2013.12.2543
[13]

Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163
[14]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[15]

Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[16]

Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567
[17]

Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026
[18]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016
[19]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[20]

Xueke Pu, Min Li. Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5149-5181. doi: 10.3934/dcdsb.2019055

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences