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December  2017, 22(10): 3903-3919. doi: 10.3934/dcdsb.2017201

On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity

Zilai Li 1,, and  Zhenhua Guo 2,

1. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
2. 

Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi'an 710069, China

* Corresponding author Zilai Li, Zhenhua Guo

Received  April 2016 Revised  July 2017 Published  August 2017

We obtain the existence of global strong solution to the free boundary problem in 1D compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when the heat conductivity depends on temperature in power law of Chapman-Enskog and the viscosity coefficient be a positive constant.

Keywords: Compressible Navier-Stokes equations, temperature-dependent heat conductivity, free boundary, global strong solution.
Mathematics Subject Classification: 35L65;35Q30;76N10.
Citation: Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201
References:
[1]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, vol. 22 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam, 1990. Translated from the Russian.  Google Scholar

[2]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[3]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990.  Google Scholar

[4]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.  Google Scholar

[5]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454.  doi: 10.1016/0362-546X(82)90058-X.  Google Scholar

[6]

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

[7]

A. Friedman, Partial Differential Equations, Krieger, New York, 1976.  Google Scholar

[8]

S. Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z., 216 (1994), 317-336.  doi: 10.1007/BF02572324.  Google Scholar

[9]

S. Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Annali di Matematica pura ed applicata, 175 (1998), 253-275.  doi: 10.1007/BF01783686.  Google Scholar

[10]

S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.  Google Scholar

[11]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coeffcients, SIAM J. Math. Anal., 42 (2010), 904-930.  doi: 10.1137/090763135.  Google Scholar

[12]

B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar

[13]

J. I. Kanel, A model system of equations for the one-dimensional motion of a gas, Differencial'nye Uravnenija, 4 (1968), 721-734.   Google Scholar

[14]

A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas, (Russian), Sibirsk. Mat. Zh., 23 (1982), 60-64.   Google Scholar

[15]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[16]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Sot. , Providence, R. I. , 1968.  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.  doi: 10.1215/kjm/1250522322.  Google Scholar

[18]

A. Matsumura and T. Nishida, The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.   Google Scholar

[19]

J. Nash, Le probleme de Cauchy pour les équations différentielles dún fluide général, Bull, Soc. Math. France, 90 (1962), 487-491.   Google Scholar

[20]

T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas nonfixed on the boundary, J. Differential Equations, 65 (1986), 49-67.  doi: 10.1016/0022-0396(86)90041-0.  Google Scholar

[21]

R. H. Pan, Global smooth solutions and the asymptotic behavior of the motion of a viscous, heat-conductive, one-dimensional real gas, J. Partial Differential Equations, 11 (1998), 273-288.   Google Scholar

[22]

R. H. Pan and W. Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.  Google Scholar

[23]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kyoto Univ., 13 (1977), 193-253.  doi: 10.2977/prims/1195190106.  Google Scholar

[24]

H. X. Liu, T. Yang, H. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal., 46 (2014), 2185-2228.  doi: 10.1137/130920617.  Google Scholar

[25]

W. G. Vincenti and C. H. Kruger, Jr. , Introduction to Physical Gas Dynamics, Physics Today, 19 (1966), p95. doi: 10.1063/1.3047788.  Google Scholar

[26]

H. Y. Wen and C. J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat-conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.  doi: 10.1137/120877829.  Google Scholar

[27]

T. Wang and H. J. Zhao, Global large solutions to a viscous heat-conducting onedimensional gas with temperature-dependent viscosity, Math. Nachr., 190 (1998), 169-183, at arXiv: 1505.05252. doi: 10.1002/mana.19981900109.  Google Scholar

[28]

Z. P. Xin, Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density, Commun, 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

show all references

References:
[1]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, vol. 22 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam, 1990. Translated from the Russian.  Google Scholar

[2]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[3]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990.  Google Scholar

[4]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.  Google Scholar

[5]

C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454.  doi: 10.1016/0362-546X(82)90058-X.  Google Scholar

[6]

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

[7]

A. Friedman, Partial Differential Equations, Krieger, New York, 1976.  Google Scholar

[8]

S. Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z., 216 (1994), 317-336.  doi: 10.1007/BF02572324.  Google Scholar

[9]

S. Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Annali di Matematica pura ed applicata, 175 (1998), 253-275.  doi: 10.1007/BF01783686.  Google Scholar

[10]

S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.  Google Scholar

[11]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coeffcients, SIAM J. Math. Anal., 42 (2010), 904-930.  doi: 10.1137/090763135.  Google Scholar

[12]

B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar

[13]

J. I. Kanel, A model system of equations for the one-dimensional motion of a gas, Differencial'nye Uravnenija, 4 (1968), 721-734.   Google Scholar

[14]

A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas, (Russian), Sibirsk. Mat. Zh., 23 (1982), 60-64.   Google Scholar

[15]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[16]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Sot. , Providence, R. I. , 1968.  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.  doi: 10.1215/kjm/1250522322.  Google Scholar

[18]

A. Matsumura and T. Nishida, The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.   Google Scholar

[19]

J. Nash, Le probleme de Cauchy pour les équations différentielles dún fluide général, Bull, Soc. Math. France, 90 (1962), 487-491.   Google Scholar

[20]

T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas nonfixed on the boundary, J. Differential Equations, 65 (1986), 49-67.  doi: 10.1016/0022-0396(86)90041-0.  Google Scholar

[21]

R. H. Pan, Global smooth solutions and the asymptotic behavior of the motion of a viscous, heat-conductive, one-dimensional real gas, J. Partial Differential Equations, 11 (1998), 273-288.   Google Scholar

[22]

R. H. Pan and W. Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.  Google Scholar

[23]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. Res. Inst. Math. Sci. Kyoto Univ., 13 (1977), 193-253.  doi: 10.2977/prims/1195190106.  Google Scholar

[24]

H. X. Liu, T. Yang, H. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal., 46 (2014), 2185-2228.  doi: 10.1137/130920617.  Google Scholar

[25]

W. G. Vincenti and C. H. Kruger, Jr. , Introduction to Physical Gas Dynamics, Physics Today, 19 (1966), p95. doi: 10.1063/1.3047788.  Google Scholar

[26]

H. Y. Wen and C. J. Zhu, Global classical large solutions to Navier-Stokes equations for viscous compressible and heat-conducting fluids with vacuum, SIAM J. Math. Anal., 45 (2013), 431-468.  doi: 10.1137/120877829.  Google Scholar

[27]

T. Wang and H. J. Zhao, Global large solutions to a viscous heat-conducting onedimensional gas with temperature-dependent viscosity, Math. Nachr., 190 (1998), 169-183, at arXiv: 1505.05252. doi: 10.1002/mana.19981900109.  Google Scholar

[28]

Z. P. Xin, Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density, Commun, 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

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