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

Previous Article
Competing interactions and traveling wave solutions in lattice differential equations
CPAA Home
This Issue
Next Article
Reaction-Diffusion equations with spatially variable exponents and large diffusion

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

Abstract
Related Papers

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).
[2]	S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases,, 3$^{rd}$ edition, (1990).
[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.
[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.
[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.
[6]	J. I. Kanel', A model system of equations for the one-dimensional motion of a gas,, \emph{Differencial$'$ nye Uravnenija}, 4 (1968), 721.
[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.
[8]	A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas,, \emph{Sibirsk. Mat. Zh.}, 23 (1982), 60.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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).
[2]	S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases,, 3$^{rd}$ edition, (1990).
[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.
[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.
[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.
[6]	J. I. Kanel', A model system of equations for the one-dimensional motion of a gas,, \emph{Differencial$'$ nye Uravnenija}, 4 (1968), 721.
[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.
[8]	A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas,, \emph{Sibirsk. Mat. Zh.}, 23 (1982), 60.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

[1]	Asim Aziz, Wasim Jamshed. Unsteady MHD slip flow of non Newtonian power-law nanofluid over a moving surface with temperature dependent thermal conductivity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 617-630. doi: 10.3934/dcdss.2018036
[2]	Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209
[3]	Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065
[4]	Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201
[5]	Jishan Fan, Fucai Li, Gen Nakamura. Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4915-4923. doi: 10.3934/dcds.2016012
[6]	Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041
[7]	Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122
[8]	Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026
[9]	Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016
[10]	P. D. Howell, J. J. Wylie, Huaxiong Huang, Robert M. Miura. Stretching of heated threads with temperature-dependent viscosity: Asymptotic analysis. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 553-572. doi: 10.3934/dcdsb.2007.7.553
[11]	Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
[12]	Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure & Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973
[13]	Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945
[14]	Katrin Grunert, Helge Holden, Xavier Raynaud. Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4209-4227. doi: 10.3934/dcds.2012.32.4209
[15]	Michael Röckner, Rongchan Zhu, Xiangchan Zhu. A remark on global solutions to random 3D vorticity equations for small initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-10. doi: 10.3934/dcdsb.2019048
[16]	Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581
[17]	Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
[18]	Zhi-Qiang Shao. Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2739-2752. doi: 10.3934/cpaa.2013.12.2739
[19]	María Teresa González Montesinos, Francisco Ortegón Gallego. The evolution thermistor problem with degenerate thermal conductivity. Communications on Pure & Applied Analysis, 2002, 1 (3) : 313-325. doi: 10.3934/cpaa.2002.1.313
[20]	María Teresa González Montesinos, Francisco Ortegón Gallego. The thermistor problem with degenerate thermal conductivity and metallic conduction. Conference Publications, 2007, 2007 (Special) : 446-455. doi: 10.3934/proc.2007.2007.446

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences