American Institute of Mathematical Sciences

Advanced
March  2016, 15(2): 477-494. doi: 10.3934/cpaa.2016.15.477

One dimensional $p$-th power Newtonian fluid with temperature-dependent thermal conductivity

Tao Wang 1,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072

Received  April 2015 Revised  November 2015 Published  January 2016

We study the initial and initial-boundary value problems for the $p$-th power Newtonian fluid in one space dimension with general large initial data. The existence and uniqueness of globally smooth non-vacuum solutions are established when the thermal conductivity is some non-negative power of the temperature. Our analysis is based on some detailed estimates on the bounds of both density and temperature.
Keywords: temperature-dependent thermal conductivity, $p$-th power Newtonian fluid, large initial data, global solutions..
Mathematics Subject Classification: Primary: 76N10; Secondary: 35A01, 35A02, 35Q3.
Citation: Tao Wang. One dimensional $p$-th power Newtonian fluid with temperature-dependent thermal conductivity. Communications on Pure and Applied Analysis, 2016, 15 (2) : 477-494. doi: 10.3934/cpaa.2016.15.477
References:
[1]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland Publishing Co., Amsterdam, 1990.

[2]

S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, 3rd edition, Cambridge University Press, Cambridge, 1990.

[3]

G. Q. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal., 23 (1992), 609-634. doi: 10.1137/0523031.

[4]

H. Cui and Z.-A. Yao, Asymptotic behavior of compressible $p$-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953. doi: 10.1016/j.jde.2014.10.011.

[5]

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

[6]

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

[7]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387.

[8]

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

[9]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Prikl. Mat. Meh., 41 (1977), 282-291.

[10]

M. Lewicka and P. B. Mucha, On temporal asymptotics for the $p$th power viscous reactive gas, Nonlinear Anal., 57 (2004), 951-969. doi: 10.1016/j.na.2003.12.001.

[11]

M. Lewicka and S. J. Watson, Temporal asymptotics for the $p$'th power Newtonian fluid in one space dimension, Z. Angew. Math. Phys., 54 (2003), 633-651. doi: 10.1007/s00033-003-1149-1.

[12]

H. Liu, T. Yang, H. Zhao and Q. 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.

[13]

R. Pan and W. 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.

[14]

Y. Qin and L. Huang, Global existence and exponential stability for the $p$th power viscous reactive gas, Nonlinear Anal., 73 (2010), 2800-2818. doi: 10.1016/j.na.2010.06.015.

[15]

Y. Qin and L. Huang, Regularity and exponential stability of the $p$th Newtonian fluid in one space dimension, Math. Models Methods Appl. Sci., 20 (2010), 589-610. doi: 10.1142/S0218202510004350.

[16]

Z. Tan, T. Yang, H. Zhao and Q. Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data, SIAM J. Math. Anal., 45 (2013), 547-571. doi: 10.1137/120876174.

show all references

References:
[1]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland Publishing Co., Amsterdam, 1990.

[2]

S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, 3rd edition, Cambridge University Press, Cambridge, 1990.

[3]

G. Q. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal., 23 (1992), 609-634. doi: 10.1137/0523031.

[4]

H. Cui and Z.-A. Yao, Asymptotic behavior of compressible $p$-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953. doi: 10.1016/j.jde.2014.10.011.

[5]

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

[6]

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

[7]

S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 384-387.

[8]

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

[9]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Prikl. Mat. Meh., 41 (1977), 282-291.

[10]

M. Lewicka and P. B. Mucha, On temporal asymptotics for the $p$th power viscous reactive gas, Nonlinear Anal., 57 (2004), 951-969. doi: 10.1016/j.na.2003.12.001.

[11]

M. Lewicka and S. J. Watson, Temporal asymptotics for the $p$'th power Newtonian fluid in one space dimension, Z. Angew. Math. Phys., 54 (2003), 633-651. doi: 10.1007/s00033-003-1149-1.

[12]

H. Liu, T. Yang, H. Zhao and Q. 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.

[13]

R. Pan and W. 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.

[14]

Y. Qin and L. Huang, Global existence and exponential stability for the $p$th power viscous reactive gas, Nonlinear Anal., 73 (2010), 2800-2818. doi: 10.1016/j.na.2010.06.015.

[15]

Y. Qin and L. Huang, Regularity and exponential stability of the $p$th Newtonian fluid in one space dimension, Math. Models Methods Appl. Sci., 20 (2010), 589-610. doi: 10.1142/S0218202510004350.

[16]

Z. Tan, T. Yang, H. Zhao and Q. Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data, SIAM J. Math. Anal., 45 (2013), 547-571. doi: 10.1137/120876174.

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