April  2021, 41(4): 1971-1999. 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  April 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (4) : 1971-1999. 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.

[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.

[3]

J.-F. CoulombelT. GoudonP. 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.

[4]

B. DucometE. 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.

[5]

L. FanL. 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.

[6]

L. FanL. 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.

[7]

W. GaoL. 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.

[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.

[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.

[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.

[11]

F. HuangJ. 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.

[12]

F. HuangA. 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.

[13]

F. HuangZ. 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.

[14]

S. JiangF. 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.

[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..

[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.

[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.

[18]

C. LattanzioC. MasciaT. NguyenR. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X.

[19]

C. LattanzioC. 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.

[20]

C. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223. 

[21]

C. LinJ.-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.

[22]

C. LinJ.-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.

[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.

[24]

R. B. LowrieJ. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.  doi: 10.1086/307515.

[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.

[26]

T. NguyenR. 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.

[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.

[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.

[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.

[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.

[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.

[32]

C. RohdeW. 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.

[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.

[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. 

[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.

[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.

[37]

W. G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York, 1965. doi: 10.1063/1.3047788.

[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.

[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.

[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.

show all references

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.

[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.

[3]

J.-F. CoulombelT. GoudonP. 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.

[4]

B. DucometE. 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.

[5]

L. FanL. 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.

[6]

L. FanL. 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.

[7]

W. GaoL. 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.

[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.

[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.

[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.

[11]

F. HuangJ. 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.

[12]

F. HuangA. 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.

[13]

F. HuangZ. 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.

[14]

S. JiangF. 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.

[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..

[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.

[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.

[18]

C. LattanzioC. MasciaT. NguyenR. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X.

[19]

C. LattanzioC. 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.

[20]

C. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223. 

[21]

C. LinJ.-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.

[22]

C. LinJ.-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.

[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.

[24]

R. B. LowrieJ. E. Morel and J. A. Hittinger, The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.  doi: 10.1086/307515.

[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.

[26]

T. NguyenR. 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.

[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.

[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.

[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.

[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.

[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.

[32]

C. RohdeW. 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.

[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.

[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. 

[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.

[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.

[37]

W. G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York, 1965. doi: 10.1063/1.3047788.

[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.

[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.

[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.

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