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

Tao Wang

doi:10.3934/cpaa.2016.15.477

Article Contents

2016, Volume 15, Issue 2: 477-494. Doi: 10.3934/cpaa.2016.15.477

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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: March 31, 2015

Revised: October 31, 2015

Published: February 2016

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Abstract

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:

$p$-th power Newtonian fluid,
temperature-dependent thermal conductivity,
large initial data,
global solutions.

Mathematics Subject Classification: Primary: 76N10; Secondary: 35A01, 35A02, 35Q35.

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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.