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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 & 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., (1990).   Google Scholar

[2]

S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases,, 3$^{rd}$ edition, (1990).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas,, \emph{Sibirsk. Mat. Zh.}, 23 (1982), 60.   Google Scholar

[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,, \emph{Prikl. Mat. Meh.}, 41 (1977), 282.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Z. Tan, T. Yang, H. Zhao and Q. Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data,, \emph{SIAM J. Math. Anal.}, 45 (2013), 547.  doi: 10.1137/120876174.  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,, North-Holland Publishing Co., (1990).   Google Scholar

[2]

S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases,, 3$^{rd}$ edition, (1990).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas,, \emph{Sibirsk. Mat. Zh.}, 23 (1982), 60.   Google Scholar

[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,, \emph{Prikl. Mat. Meh.}, 41 (1977), 282.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

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