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September  2016, 9(3): 469-514. doi: 10.3934/krm.2016004

Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients

 1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 2 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

Received  January 2016 Revised  February 2016 Published  May 2016

We study the nonlinear stability of rarefaction waves to the Cauchy problem of one-dimensional compressible Navier-Stokes equations for a viscous and heat conducting ideal polytropic gas when the transport coefficients depend on both temperature and density. When the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially, we show that rarefaction waves are nonlinear stable provided that $(\gamma- 1)\cdot H^3(\mathbb{R})$-norm of the initial perturbation is suitably small with $\gamma>1$ being the adiabatic gas constant.
Citation: Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic and Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
References:
 [1] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990. [2] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Appl. Math. Sci. 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [3] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454. doi: 10.1016/0362-546X(82)90058-X. [4] R. Duan, H.-X. Liu and H.-J. Zhao, Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation, Trans. Amer. Math. Soc., 361 (2009), 453-493. doi: 10.1090/S0002-9947-08-04637-0. [5] H. Grad, Asymptotic theory of the Boltzmann equation II, in rarefied gas dynamics, J.A. Laurmann, ed., Academic Press, New York, 1963, pp. 26-59. [6] L. Hsiao and S. Jiang, Nonlinear hyperbolic-parabolic coupled systems, In: Handbook of Differential Equations, Vol. 1: Evolutionary Equations. Chapter 4, pp. 287-384. Elsevier, 2004. [7] F.-M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116. doi: 10.1007/s00205-009-0267-0. [8] F.-M. Huang and A. Matsumura, Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation, Comm. Math. Phys., 289 (2009), 841-861. doi: 10.1007/s00220-009-0843-z. [9] F.-M. Huang, A. Matsumura and Z.-P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 179 (2006), 55-77. doi: 10.1007/s00205-005-0380-7. [10] F.-M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Preprint at arXiv: 1502.00211. [11] F.-M. Huang, Z.-P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014. [12] S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193. doi: 10.1007/s002200050526. [13] S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Monogr. Surv. Pure Appl. Math., 112, Chapman & Hall/CRC, Boca Raton, FL, 2000. [14] S. Jiang and P. Zhang, Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas, Quart. Appl. Math., 61 (2003), 435-449. [15] 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. [16] Y. Kanel', On a model system of equations of one-dimensional gas motion, Differ. Uravn., 4 (1968), pp. 374-380. [17] S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. [18] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behaviour of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. A., 62 (1986), 249-252. doi: 10.3792/pjaa.62.249. [19] S. Kawashima and T. Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ., 21 (1983), 825-837. [20] B. Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103. doi: 10.1016/0022-0396(85)90023-3. [21] A. V. Kazhikhov, Correctness "in the whole" of the mixed boundary value problems for a model system of equations of a viscous gas (in Russian), Dinamika Sploshn. Sredy, 21 (1975), 18-47. [22] 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, J. Appl. Math. Mech., 41 (1977), 273-282. [23] J. Li and Z. Liang, Some uniform estimates and large-time behavior for one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208, arXiv:1404.2214. doi: 10.1007/s00205-015-0952-0. [24] H.-X. Liu, T. Yang, H.-J. Zhao and Q.-Y. 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. [25] T.-P. Liu, Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J., 26 (1977), 147-177. doi: 10.1512/iumj.1977.26.26011. [26] T.-P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Commun. Pure Appl. Math., 39 (1986), 565-594. doi: 10.1002/cpa.3160390502. [27] T.-P. Liu and Z.-P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465. doi: 10.1007/BF01466726. [28] T.-P. Liu and Z.-P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84. doi: 10.4310/AJM.1997.v1.n1.a3. [29] T.-P. Liu and Y.-N. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp. doi: 10.1090/memo/0599. [30] T.-P. Liu and Y.-N. Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 234 (2015), vi+168 pp. doi: 10.1090/memo/1105. [31] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Proceedings of the Third World Congress of Nonlinear Analysts, Part 6 (Catania, 2000), Nonlinear Anal., 47 (2001), 4269-4282. doi: 10.1016/S0362-546X(01)00542-9. [32] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [33] A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25. doi: 10.1007/BF03167036. [34] A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13. doi: 10.1007/BF03167088. [35] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335. doi: 10.1007/BF02101095. [36] A. Matsumura and K. Nishihara, Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect, Quart. Appl. Math., 58 (2000), 69-83. [37] T. Nishida and J. A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math., 26 (1973), 183-200. doi: 10.1002/cpa.3160260205. [38] K. Nishihara, T. Yang and H.-J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X. [39] M. Okada and S. Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ., 23 (1983), 55-71. [40] R.-H. Pan and W.-Z. 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. [41] Z. Tan, T. Yang, H.-J. Zhao and Q.-Y. 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. [42] J. B. Temple, Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics, J. Differential Equations, 41 (1981), 96-161. doi: 10.1016/0022-0396(81)90055-3. [43] W. G. Vincenti and C. H. Kruger, Introduction to Physical Gas Dynamics, Cambridge Math. Lib., Cambridge University Press, Cambridge, UK, 1975. [44] L. Wan, T. Wang and H.-J. Zhao, Asymptotic stability of wave patterns to compressible viscous and heat-conducting gases in the half space,, , (). [45] L. Wan, T. Wang and Q.-Y. Zou, Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation, IOP Publishing Ltd & London Mathematical Society, 49 (2016), arXiv:1503.03922. doi: 10.1088/0951-7715/29/4/1329. [46] T. Wang and H.-J. Zhao, Global large solutions to a viscous heat-conducting one-dimensional gas with temperature-dependent viscosity, Preprint at arXiv:1505.05252 (2015). [47] T. Wang, H.-J. Zhao and Q.-Y. Zou, One-dimensional compressible Navier-Stokes equations with large density oscillation, Kinet. Relat. Models, 6 (2013), 649-670. doi: 10.3934/krm.2013.6.649. [48] Y. B. Zel'dovich and Y. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, Vol. II, Academic Press, New York, 1967. [49] A. A. Zlotnik and A. A. Amosov, On the stability of generalized solutions of equations of one-dimensional motion of a viscous heat-conducting gas, Siberian Math. J., 38 (1997), 663-684. doi: 10.1007/BF02674573.

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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, Amsterdam, 1990. [2] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Appl. Math. Sci. 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [3] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454. doi: 10.1016/0362-546X(82)90058-X. [4] R. Duan, H.-X. Liu and H.-J. Zhao, Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation, Trans. Amer. Math. Soc., 361 (2009), 453-493. doi: 10.1090/S0002-9947-08-04637-0. [5] H. Grad, Asymptotic theory of the Boltzmann equation II, in rarefied gas dynamics, J.A. Laurmann, ed., Academic Press, New York, 1963, pp. 26-59. [6] L. Hsiao and S. Jiang, Nonlinear hyperbolic-parabolic coupled systems, In: Handbook of Differential Equations, Vol. 1: Evolutionary Equations. Chapter 4, pp. 287-384. Elsevier, 2004. [7] F.-M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116. doi: 10.1007/s00205-009-0267-0. [8] F.-M. Huang and A. Matsumura, Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation, Comm. Math. Phys., 289 (2009), 841-861. doi: 10.1007/s00220-009-0843-z. [9] F.-M. Huang, A. Matsumura and Z.-P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 179 (2006), 55-77. doi: 10.1007/s00205-005-0380-7. [10] F.-M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Preprint at arXiv: 1502.00211. [11] F.-M. Huang, Z.-P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014. [12] S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193. doi: 10.1007/s002200050526. [13] S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Monogr. Surv. Pure Appl. Math., 112, Chapman & Hall/CRC, Boca Raton, FL, 2000. [14] S. Jiang and P. Zhang, Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas, Quart. Appl. Math., 61 (2003), 435-449. [15] 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. [16] Y. Kanel', On a model system of equations of one-dimensional gas motion, Differ. Uravn., 4 (1968), pp. 374-380. [17] S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. [18] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behaviour of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. A., 62 (1986), 249-252. doi: 10.3792/pjaa.62.249. [19] S. Kawashima and T. Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ., 21 (1983), 825-837. [20] B. Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103. doi: 10.1016/0022-0396(85)90023-3. [21] A. V. Kazhikhov, Correctness "in the whole" of the mixed boundary value problems for a model system of equations of a viscous gas (in Russian), Dinamika Sploshn. Sredy, 21 (1975), 18-47. [22] 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, J. Appl. Math. Mech., 41 (1977), 273-282. [23] J. Li and Z. Liang, Some uniform estimates and large-time behavior for one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208, arXiv:1404.2214. doi: 10.1007/s00205-015-0952-0. [24] H.-X. Liu, T. Yang, H.-J. Zhao and Q.-Y. 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. [25] T.-P. Liu, Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J., 26 (1977), 147-177. doi: 10.1512/iumj.1977.26.26011. [26] T.-P. Liu, Shock waves for compressible Navier-Stokes equations are stable, Commun. Pure Appl. Math., 39 (1986), 565-594. doi: 10.1002/cpa.3160390502. [27] T.-P. Liu and Z.-P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465. doi: 10.1007/BF01466726. [28] T.-P. Liu and Z.-P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84. doi: 10.4310/AJM.1997.v1.n1.a3. [29] T.-P. Liu and Y.-N. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp. doi: 10.1090/memo/0599. [30] T.-P. Liu and Y.-N. Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 234 (2015), vi+168 pp. doi: 10.1090/memo/1105. [31] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Proceedings of the Third World Congress of Nonlinear Analysts, Part 6 (Catania, 2000), Nonlinear Anal., 47 (2001), 4269-4282. doi: 10.1016/S0362-546X(01)00542-9. [32] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [33] A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25. doi: 10.1007/BF03167036. [34] A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13. doi: 10.1007/BF03167088. [35] A. Matsumura and K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335. doi: 10.1007/BF02101095. [36] A. Matsumura and K. Nishihara, Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect, Quart. Appl. Math., 58 (2000), 69-83. [37] T. Nishida and J. A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math., 26 (1973), 183-200. doi: 10.1002/cpa.3160260205. [38] K. Nishihara, T. Yang and H.-J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X. [39] M. Okada and S. Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ., 23 (1983), 55-71. [40] R.-H. Pan and W.-Z. 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. [41] Z. Tan, T. Yang, H.-J. Zhao and Q.-Y. 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. [42] J. B. Temple, Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics, J. Differential Equations, 41 (1981), 96-161. doi: 10.1016/0022-0396(81)90055-3. [43] W. G. Vincenti and C. H. Kruger, Introduction to Physical Gas Dynamics, Cambridge Math. Lib., Cambridge University Press, Cambridge, UK, 1975. [44] L. Wan, T. Wang and H.-J. Zhao, Asymptotic stability of wave patterns to compressible viscous and heat-conducting gases in the half space,, , (). [45] L. Wan, T. Wang and Q.-Y. Zou, Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation, IOP Publishing Ltd & London Mathematical Society, 49 (2016), arXiv:1503.03922. doi: 10.1088/0951-7715/29/4/1329. [46] T. Wang and H.-J. Zhao, Global large solutions to a viscous heat-conducting one-dimensional gas with temperature-dependent viscosity, Preprint at arXiv:1505.05252 (2015). [47] T. Wang, H.-J. Zhao and Q.-Y. Zou, One-dimensional compressible Navier-Stokes equations with large density oscillation, Kinet. Relat. Models, 6 (2013), 649-670. doi: 10.3934/krm.2013.6.649. [48] Y. B. Zel'dovich and Y. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, Vol. II, Academic Press, New York, 1967. [49] A. A. Zlotnik and A. A. Amosov, On the stability of generalized solutions of equations of one-dimensional motion of a viscous heat-conducting gas, Siberian Math. J., 38 (1997), 663-684. doi: 10.1007/BF02674573.
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