# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020349

## Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations

 1 School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China 2 Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China 3 Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China

* Corresponding author: Lizhi Ruan, rlz@mail.ccnu.edu.cn

Received  March 2020 Revised  August 2020 Published  October 2020

This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [6], since the rarefaction wave is a nonlinear expansive wave, while the contact discontinuity wave is a linearly degenerate diffusive wave. So we need to take good advantage of properties of the viscous contact discontinuity wave instead. Moreover, series of tricky argument on the boundary is done carefully based on the construction and the properties of the viscous contact discontinuity wave for the radiative Euler equations. Our result shows that radiation contributes to the stabilization effect for the supersonic inflow problem.

Citation: Lili Fan, Lizhi Ruan, Wei Xiang. Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020349
##### References:
 [1] A. M. Blokhin and Yu. L. Trakhinin, Shock-wave stability for one model of radiation hydrodynamics, J. Appl. Mech. Tech. Phys., 37 (1996), 775-784.  doi: 10.1007/BF02369253.  Google Scholar [2] C. Buet and B. Despres, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418.  doi: 10.1016/S0022-4073(03)00233-4.  Google Scholar [3] J.-F. Coulombel, T. Goudon, P. Lafitte and C. Lin, Analysis of large amplitude shock profiles for non-equilibrium radiative hydrodynamics: Formation of Zeldovich spikes, Shock Waves, 22 (2012), 181-197.  doi: 10.1007/s00193-012-0368-9.  Google Scholar [4] B. Ducomet, E. Feireisl and S. Nečasova, On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 797-812.  doi: 10.1016/j.anihpc.2011.06.002.  Google Scholar [5] L. Fan, L. Ruan and W. Xiang, Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1-25.  doi: 10.1016/j.anihpc.2018.03.008.  Google Scholar [6] L. Fan, L. Ruan and W. Xiang, Asymptotic stability of rarefaction wave for the inflow problem governed by the one-dimensional radiative Euler equations, SIAM J. Math. Anal., 51 (2019), 595-625.  doi: 10.1137/18M1203043.  Google Scholar [7] W. Gao, L. Ruan and C. Zhu, Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$-dimensions, J. Differential Equations, 244 (2008), 2614-2640.  doi: 10.1016/j.jde.2008.02.023.  Google Scholar [8] P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Simul., 4 (2005), 1245-1279.  doi: 10.1137/040621041.  Google Scholar [9] H. Hong, Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations, J. Differential Equations, 252 (2012), 3482-3505.  doi: 10.1016/j.jde.2011.11.015.  Google Scholar [10] F. Huang and X. Li, Convergence to the rarefaction wave for a model of radiating gas in one-dimension, Acta Math. Appl. Sin. Engl. Ser., 32 (2016), 239-256.  doi: 10.1007/s10255-016-0576-7.  Google Scholar [11] F. 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.  Google Scholar [12] F. Huang, A. Matsumura and Z. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7.  Google Scholar [13] F. Huang, Z. 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.  Google Scholar [14] S. Jiang, F. Li and F. Xie, Nonrelativistic limit of the compressible Navier-Stokes-Fourier-P1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.  doi: 10.1137/140987596.  Google Scholar [15] S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, Analysis of Systems of Conservation Laws (Aachen, 1997), Chapman Hall/CRC Monogr. Surv. Pure. Appl. Math., Chapman Hall/CRC, Boca Raton, FL, 99 1999, 87–127..  Google Scholar [16] S. Kawashima and S. Nishibata, Shock waves for a model system of a radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117.  doi: 10.1137/S0036141097322169.  Google Scholar [17] S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250.  doi: 10.2206/kyushujm.58.211.  Google Scholar [18] C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X.  Google Scholar [19] C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.  doi: 10.1512/iumj.2007.56.3043.  Google Scholar [20] C. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223.   Google Scholar [21] C. Lin, J.-F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gas, Phys. D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012.  Google Scholar [22] C. Lin, J.-F. Coulombel and T. Goudon, Asymptotic stability of shock profiles in radiative hydrodynamics, C. R. Math. Acad. Sci. Paris, 345 (2007), 625-628.  doi: 10.1016/j.crma.2007.10.029.  Google Scholar [23] T.-P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796.  doi: 10.1002/cpa.3160300605.  Google Scholar [24] R. B. Lowrie, J. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.  doi: 10.1086/307515.  Google Scholar [25] C. Mascia, Small, medium and large shock waves for radiative Euler equations, Phys. D, 245 (2013), 46-56.  doi: 10.1016/j.physd.2012.11.008.  Google Scholar [26] T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453.  doi: 10.1016/j.physd.2010.01.011.  Google Scholar [27] M. Nishikawa and S. Nishibata, Convergence rates toward the travelling waves for a model system of the radiating gas, Math. Methods Appl. Sci., 30 (2007), 649-663.  doi: 10.1002/mma.800.  Google Scholar [28] M. Ohnawa, Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities, Kinet. Relat. Models, 5 (2012), 857-872.  doi: 10.3934/krm.2012.5.857.  Google Scholar [29] M. Ohnawa, $L^\infty$-stability of continuous shock waves in a radiating gas model, SIAM J. Math. Anal., 46 (2014), 2136-2159.  doi: 10.1137/130935252.  Google Scholar [30] X. Qin and Y. Wang, Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 41 (2009), 2057-2087.  doi: 10.1137/09075425X.  Google Scholar [31] X. Qin and Y. Wang, Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 43 (2011), 341-366.  doi: 10.1137/100793463.  Google Scholar [32] C. Rohde, W. Wang and F. Xie, Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: Superposition of rarefaction and contact waves, Commun. Pure Appl. Anal., 12 (2013), 2145-2171.  doi: 10.3934/cpaa.2013.12.2145.  Google Scholar [33] L. Ruan and J. Zhang, Asymptotic stability of rarefaction wave for hyperbolic-elliptic coupled system in radiating gas, Acta Math. Sci. Ser. B Engl. Ed., 27 (2007), 347-360.  doi: 10.1016/S0252-9602(07)60035-6.  Google Scholar [34] L. Ruan and C. Zhu, Asymptotic decay toward rarefaction wave for a hyperbolic-elliptic coupled system on half space, J. Partial Differential Equations, 21 (2008), 173-192.   Google Scholar [35] L. Ruan and C. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas, J. Differential Equations, 249 (2010), 2076-2110.  doi: 10.1016/j.jde.2010.07.029.  Google Scholar [36] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [37] W. G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York, 1965. doi: 10.1063/1.3047788.  Google Scholar [38] J. Wang and F. Xie, Singular limit to strong contact discontinuity for a 1D compressible radiation hydrodynamics model, SIAM J. Math. Anal., 43 (2011), 1189-1204.  doi: 10.1137/100792792.  Google Scholar [39] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system, J. Differential Equations, 251 (2011), 1030-1055.  doi: 10.1016/j.jde.2011.03.011.  Google Scholar [40] F. Xie, Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1075-1100.  doi: 10.3934/dcdsb.2012.17.1075.  Google Scholar

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##### References:
 [1] A. M. Blokhin and Yu. L. Trakhinin, Shock-wave stability for one model of radiation hydrodynamics, J. Appl. Mech. Tech. Phys., 37 (1996), 775-784.  doi: 10.1007/BF02369253.  Google Scholar [2] C. Buet and B. Despres, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418.  doi: 10.1016/S0022-4073(03)00233-4.  Google Scholar [3] J.-F. Coulombel, T. Goudon, P. Lafitte and C. Lin, Analysis of large amplitude shock profiles for non-equilibrium radiative hydrodynamics: Formation of Zeldovich spikes, Shock Waves, 22 (2012), 181-197.  doi: 10.1007/s00193-012-0368-9.  Google Scholar [4] B. Ducomet, E. Feireisl and S. Nečasova, On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 797-812.  doi: 10.1016/j.anihpc.2011.06.002.  Google Scholar [5] L. Fan, L. Ruan and W. Xiang, Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1-25.  doi: 10.1016/j.anihpc.2018.03.008.  Google Scholar [6] L. Fan, L. Ruan and W. Xiang, Asymptotic stability of rarefaction wave for the inflow problem governed by the one-dimensional radiative Euler equations, SIAM J. Math. Anal., 51 (2019), 595-625.  doi: 10.1137/18M1203043.  Google Scholar [7] W. Gao, L. Ruan and C. Zhu, Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$-dimensions, J. Differential Equations, 244 (2008), 2614-2640.  doi: 10.1016/j.jde.2008.02.023.  Google Scholar [8] P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Simul., 4 (2005), 1245-1279.  doi: 10.1137/040621041.  Google Scholar [9] H. Hong, Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations, J. Differential Equations, 252 (2012), 3482-3505.  doi: 10.1016/j.jde.2011.11.015.  Google Scholar [10] F. Huang and X. Li, Convergence to the rarefaction wave for a model of radiating gas in one-dimension, Acta Math. Appl. Sin. Engl. Ser., 32 (2016), 239-256.  doi: 10.1007/s10255-016-0576-7.  Google Scholar [11] F. 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.  Google Scholar [12] F. Huang, A. Matsumura and Z. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77.  doi: 10.1007/s00205-005-0380-7.  Google Scholar [13] F. Huang, Z. 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.  Google Scholar [14] S. Jiang, F. Li and F. Xie, Nonrelativistic limit of the compressible Navier-Stokes-Fourier-P1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.  doi: 10.1137/140987596.  Google Scholar [15] S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, Analysis of Systems of Conservation Laws (Aachen, 1997), Chapman Hall/CRC Monogr. Surv. Pure. Appl. Math., Chapman Hall/CRC, Boca Raton, FL, 99 1999, 87–127..  Google Scholar [16] S. Kawashima and S. Nishibata, Shock waves for a model system of a radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117.  doi: 10.1137/S0036141097322169.  Google Scholar [17] S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250.  doi: 10.2206/kyushujm.58.211.  Google Scholar [18] C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X.  Google Scholar [19] C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.  doi: 10.1512/iumj.2007.56.3043.  Google Scholar [20] C. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223.   Google Scholar [21] C. Lin, J.-F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gas, Phys. D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012.  Google Scholar [22] C. Lin, J.-F. Coulombel and T. Goudon, Asymptotic stability of shock profiles in radiative hydrodynamics, C. R. Math. Acad. Sci. Paris, 345 (2007), 625-628.  doi: 10.1016/j.crma.2007.10.029.  Google Scholar [23] T.-P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796.  doi: 10.1002/cpa.3160300605.  Google Scholar [24] R. B. Lowrie, J. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.  doi: 10.1086/307515.  Google Scholar [25] C. Mascia, Small, medium and large shock waves for radiative Euler equations, Phys. D, 245 (2013), 46-56.  doi: 10.1016/j.physd.2012.11.008.  Google Scholar [26] T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453.  doi: 10.1016/j.physd.2010.01.011.  Google Scholar [27] M. Nishikawa and S. Nishibata, Convergence rates toward the travelling waves for a model system of the radiating gas, Math. Methods Appl. Sci., 30 (2007), 649-663.  doi: 10.1002/mma.800.  Google Scholar [28] M. Ohnawa, Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities, Kinet. Relat. Models, 5 (2012), 857-872.  doi: 10.3934/krm.2012.5.857.  Google Scholar [29] M. Ohnawa, $L^\infty$-stability of continuous shock waves in a radiating gas model, SIAM J. Math. Anal., 46 (2014), 2136-2159.  doi: 10.1137/130935252.  Google Scholar [30] X. Qin and Y. Wang, Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 41 (2009), 2057-2087.  doi: 10.1137/09075425X.  Google Scholar [31] X. Qin and Y. Wang, Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 43 (2011), 341-366.  doi: 10.1137/100793463.  Google Scholar [32] C. Rohde, W. Wang and F. Xie, Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: Superposition of rarefaction and contact waves, Commun. Pure Appl. Anal., 12 (2013), 2145-2171.  doi: 10.3934/cpaa.2013.12.2145.  Google Scholar [33] L. Ruan and J. Zhang, Asymptotic stability of rarefaction wave for hyperbolic-elliptic coupled system in radiating gas, Acta Math. Sci. Ser. B Engl. Ed., 27 (2007), 347-360.  doi: 10.1016/S0252-9602(07)60035-6.  Google Scholar [34] L. Ruan and C. Zhu, Asymptotic decay toward rarefaction wave for a hyperbolic-elliptic coupled system on half space, J. Partial Differential Equations, 21 (2008), 173-192.   Google Scholar [35] L. Ruan and C. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas, J. Differential Equations, 249 (2010), 2076-2110.  doi: 10.1016/j.jde.2010.07.029.  Google Scholar [36] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [37] W. G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York, 1965. doi: 10.1063/1.3047788.  Google Scholar [38] J. Wang and F. Xie, Singular limit to strong contact discontinuity for a 1D compressible radiation hydrodynamics model, SIAM J. Math. Anal., 43 (2011), 1189-1204.  doi: 10.1137/100792792.  Google Scholar [39] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system, J. Differential Equations, 251 (2011), 1030-1055.  doi: 10.1016/j.jde.2011.03.011.  Google Scholar [40] F. Xie, Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1075-1100.  doi: 10.3934/dcdsb.2012.17.1075.  Google Scholar
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