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Separated characteristics and global solvability for the one and one-half dimensional Vlasov Maxwell system
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 |
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. |
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, 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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