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On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
E. Feireisl,
Dynamics of Viscous Compressible Fluids, Oxford Univ. Press, Oxford, 2004. |
[7] |
A. Friedman,
Partial Differential Equations, Krieger, New York, 1976. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. I. Kanel,
A model system of equations for the one-dimensional motion of a gas, Differencial'nye Uravnenija, 4 (1968), 721-734.
|
[14] |
A. V. Kazhikhov,
On the Cauchy problem for the equations of a viscous gas, (Russian), Sibirsk. Mat. Zh., 23 (1982), 60-64.
|
[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. |
[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. |
[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. |
[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.
|
[19] |
J. Nash,
Le probleme de Cauchy pour les équations différentielles dún fluide général, Bull, Soc. Math. France, 90 (1962), 487-491.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[25] |
W. G. Vincenti and C. H. Kruger, Jr. ,
Introduction to Physical Gas Dynamics,
Physics Today, 19 (1966), p95.
doi: 10.1063/1.3047788. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
E. Feireisl,
Dynamics of Viscous Compressible Fluids, Oxford Univ. Press, Oxford, 2004. |
[7] |
A. Friedman,
Partial Differential Equations, Krieger, New York, 1976. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. I. Kanel,
A model system of equations for the one-dimensional motion of a gas, Differencial'nye Uravnenija, 4 (1968), 721-734.
|
[14] |
A. V. Kazhikhov,
On the Cauchy problem for the equations of a viscous gas, (Russian), Sibirsk. Mat. Zh., 23 (1982), 60-64.
|
[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. |
[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. |
[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. |
[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.
|
[19] |
J. Nash,
Le probleme de Cauchy pour les équations différentielles dún fluide général, Bull, Soc. Math. France, 90 (1962), 487-491.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[25] |
W. G. Vincenti and C. H. Kruger, Jr. ,
Introduction to Physical Gas Dynamics,
Physics Today, 19 (1966), p95.
doi: 10.1063/1.3047788. |
[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. |
[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. |
[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. |
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