# American Institute of Mathematical Sciences

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

 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.
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., Amsterdam, 1990.  Google Scholar [2] S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, 3rd edition, Cambridge University Press, Cambridge, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] 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 [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.  Google Scholar [8] A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas, Sibirsk. Mat. Zh., 23 (1982), 60-64.  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, Prikl. Mat. Meh., 41 (1977), 282-291.  Google Scholar [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.  Google Scholar [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.  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, SIAM J. Math. Anal., 46 (2014), 2185-2228. doi: 10.1137/130920617.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, SIAM J. Math. Anal., 45 (2013), 547-571. doi: 10.1137/120876174.  Google Scholar

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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.  Google Scholar [2] S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, 3rd edition, Cambridge University Press, Cambridge, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] 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 [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.  Google Scholar [8] A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas, Sibirsk. Mat. Zh., 23 (1982), 60-64.  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, Prikl. Mat. Meh., 41 (1977), 282-291.  Google Scholar [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.  Google Scholar [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.  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, SIAM J. Math. Anal., 46 (2014), 2185-2228. doi: 10.1137/130920617.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, SIAM J. Math. Anal., 45 (2013), 547-571. doi: 10.1137/120876174.  Google Scholar
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