Steady asymmetric vortex pairs for Euler equations
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
Lili Fan 1, , Lizhi Ruan 2,, and Wei Xiang 3,
School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China
Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
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
Full Text(HTML)
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.
Keywords: Radiative Euler equations, inflow problem, viscous contact wave, asymptotic stability, supersonic.
Mathematics Subject Classification: Primary: 35B35, 35B40, 35M30, 35Q35, 76N10; Secondary: 76N15.
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, 2021, 41 (4) : 1971-1999. doi: 10.3934/dcds.2020349
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
C. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223. Google Scholar
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
W. G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York, 1965. doi: 10.1063/1.3047788. Google Scholar
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
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
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
Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062
Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075
Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control & Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010
La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981
Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic & Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008
Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419
Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409
Yanbo Hu, Tong Li. Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations. Kinetic & Related Models, 2019, 12 (6) : 1197-1228. doi: 10.3934/krm.2019046
Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198
Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569
Cristian A. Coclici, Jörg Heiermann, Gh. Moroşanu, W. L. Wendland. Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 137-163. doi: 10.3934/dcds.2004.10.137
Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic & Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
J-F. Clouët, R. Sentis. Milne problem for non-grey radiative transfer. Kinetic & Related Models, 2009, 2 (2) : 345-362. doi: 10.3934/krm.2009.2.345
Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176
Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127
Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361
Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469
PDF downloads (202)
HTML views (162)
Lili Fan Lizhi Ruan Wei Xiang
Article outline