-
Previous Article
Long-time stability of small FPU solitary waves
- DCDS Home
- This Issue
-
Next Article
Performance bounds for the mean-field limit of constrained dynamics
Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics
1. | Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China |
2. | Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA |
We are concerned with the global well-posedness of the diffusion approximation model in radiation hydrodynamics, which describe the compressible fluid dynamics taking into account the radiation effect under the non-local thermal equilibrium case. The model consist of the compressible Navier-Stokes equations coupled with the radiative transport equation with non-local terms. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution is proved based on the classical energy estimates, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between photons and matter.
References:
[1] |
J. W. Bond, K. M. Watson and J. A. Welch, Atomic Theory of Gas Dynamics, Addison-Wesley, Rewading, Massachusetts, 1965. Google Scholar |
[2] |
C. Buet and B. Després,
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. A. Carrillo, R. Duan and A. Moussa,
Global classical solution close to equillibrium to the Vlasov-Euler-Fokker-Planck system, Kinet. Relat. Model, 4 (2011), 227-258.
doi: 10.3934/krm.2011.4.227. |
[4] |
J. I. Castor,
Radiation Hydrodynamics Cambridge University Press, 2004.
doi: 10.1017/CBO9780511536182. |
[5] |
B. Ducomet and E. Feireisl,
The equation of magnetohydrodynamics: On the interation between matter and radiation in the evlution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
[6] |
B. Ducomet, E. Feireisl and S. Necasova,
On a model in radiation hydrodynamics, Ann. Inst. H. Poincar Anal. Non Linaire, 28 (2011), 797-812.
doi: 10.1016/j.anihpc.2011.06.002. |
[7] |
Th. Goudon and P. Lafitte,
A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, SIAM Multiscale Model. Simul., 4 (2005), 1245-1279.
doi: 10.1137/040621041. |
[8] |
X. Hu and D. Wang,
Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.
doi: 10.1016/j.jde.2010.10.017. |
[9] |
X. Hu and D. Wang,
Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equation, 252 (2012), 4027-4067.
doi: 10.1016/j.jde.2011.11.021. |
[10] |
D. Hoff and K. Zumbrum,
Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. J, 44 (1995), 603-676.
doi: 10.1512/iumj.1995.44.2003. |
[11] |
E. Hopf,
Mathematical Problems of Radiative Equilibrium, Stechert-Hafner, New York, 1964. |
[12] |
S. Jiang, F. C. Li and F. Xie,
Nonrelativistic limits of the compressible Navier-Stokes-FourierP1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.
doi: 10.1137/140987596. |
[13] |
S. Jiang, F. Xie and J. W. Zhang, A global existence result in radiation hydrodynamics,
Industrial and Applied Mathematics in China, Series in Contemporary Applied Mathematics,
High Edu. Press and World Scientific. Beijing, Singapore, 10 (2009), 25–48. |
[14] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984. Google Scholar |
[15] |
S. Kawashima and S. Nishibata,
A singular limit for hyperbolic-elliptic coupled systems in
radiation hydrodynamics, Indiana Univ. Math. J., 50 (2001), 567-589.
doi: 10.1512/iumj.2001.50.1797. |
[16] |
S. Kawashima and S. Nishibata,
Shock waves for a model system of the radiating gases, SIAM J. Math. Anal., 30 (1999), 95-117.
doi: 10.1137/S0036141097322169. |
[17] |
R. Kippenhahn and A. Weigert, Stellar Structure and Evolution Springer Verlag, Berlin-Heidelberg, 1994. Google Scholar |
[18] |
S. Klainerman and A. Majda,
Singular limits of quasilinear hyperbolic systems with large
parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math, 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[19] |
C. Lin,
Asymptotic stability of rarefaction waves in radiation hydrodynamics, Comm. Math. Sci., 9 (2011), 207-223.
doi: 10.4310/CMS.2011.v9.n1.a10. |
[20] |
C. Lin, J. F. Coulombel and Th. Goudon,
Shock profiles for non equilibrium radiating gases, Physica D, 218 (2006), 83-94.
doi: 10.1016/j.physd.2006.04.012. |
[21] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser. A, 55 (1979), 337-342.
doi: 10.3792/pjaa.55.337. |
[22] |
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.
|
[23] |
D. Mihalas and B. Weibel-Mihalas,
Foundations of Radiation Hydrodynamics, Oxford University Press, 1984. |
[24] |
S. S. Penner and D. B. Olfe, Radiation and Reentry, Academic Press, New York, 1968. Google Scholar |
[25] |
G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1973. Google Scholar |
[26] |
C. Rohde and W.-A. Yong,
The nonrelativistic limit in radiation hydrodynamics: I. Weak
entropy solutions for a model problem, J. Diff. Eqns., 234 (2007), 91-109.
doi: 10.1016/j.jde.2006.11.010. |
[27] |
R. N. Thomas, Some Aspects of Non-Equilibrium Thermodynamics in the Presence of a Radiation Field, University of Colorado Press, Boulder, Colorado, 1965. Google Scholar |
[28] |
W. J. Wang and F. Xie,
The initial value problem for a multi-dimensional radiation hydrodynamics model with viscosity,, Math. Methods Appl. Sci., 34 (2011), 776-791.
doi: 10.1002/mma.1398. |
[29] |
Y. B. Zeldovich and Y. P. Raizer, Phsics of Shock Waves and High-Temperture Hydrodynamic Phenomenon, Academic Press, 1966. Google Scholar |
[30] |
X. Zhong and S. Jiang,
Local existence and finite time blow-up in multidimensional radiation
hydrodynamics,, J. Math. Fluid Mech., 9 (2007), 543-564.
doi: 10.1007/s00021-005-0213-3. |
show all references
References:
[1] |
J. W. Bond, K. M. Watson and J. A. Welch, Atomic Theory of Gas Dynamics, Addison-Wesley, Rewading, Massachusetts, 1965. Google Scholar |
[2] |
C. Buet and B. Després,
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. A. Carrillo, R. Duan and A. Moussa,
Global classical solution close to equillibrium to the Vlasov-Euler-Fokker-Planck system, Kinet. Relat. Model, 4 (2011), 227-258.
doi: 10.3934/krm.2011.4.227. |
[4] |
J. I. Castor,
Radiation Hydrodynamics Cambridge University Press, 2004.
doi: 10.1017/CBO9780511536182. |
[5] |
B. Ducomet and E. Feireisl,
The equation of magnetohydrodynamics: On the interation between matter and radiation in the evlution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.
doi: 10.1007/s00220-006-0052-y. |
[6] |
B. Ducomet, E. Feireisl and S. Necasova,
On a model in radiation hydrodynamics, Ann. Inst. H. Poincar Anal. Non Linaire, 28 (2011), 797-812.
doi: 10.1016/j.anihpc.2011.06.002. |
[7] |
Th. Goudon and P. Lafitte,
A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, SIAM Multiscale Model. Simul., 4 (2005), 1245-1279.
doi: 10.1137/040621041. |
[8] |
X. Hu and D. Wang,
Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.
doi: 10.1016/j.jde.2010.10.017. |
[9] |
X. Hu and D. Wang,
Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equation, 252 (2012), 4027-4067.
doi: 10.1016/j.jde.2011.11.021. |
[10] |
D. Hoff and K. Zumbrum,
Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. J, 44 (1995), 603-676.
doi: 10.1512/iumj.1995.44.2003. |
[11] |
E. Hopf,
Mathematical Problems of Radiative Equilibrium, Stechert-Hafner, New York, 1964. |
[12] |
S. Jiang, F. C. Li and F. Xie,
Nonrelativistic limits of the compressible Navier-Stokes-FourierP1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.
doi: 10.1137/140987596. |
[13] |
S. Jiang, F. Xie and J. W. Zhang, A global existence result in radiation hydrodynamics,
Industrial and Applied Mathematics in China, Series in Contemporary Applied Mathematics,
High Edu. Press and World Scientific. Beijing, Singapore, 10 (2009), 25–48. |
[14] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984. Google Scholar |
[15] |
S. Kawashima and S. Nishibata,
A singular limit for hyperbolic-elliptic coupled systems in
radiation hydrodynamics, Indiana Univ. Math. J., 50 (2001), 567-589.
doi: 10.1512/iumj.2001.50.1797. |
[16] |
S. Kawashima and S. Nishibata,
Shock waves for a model system of the radiating gases, SIAM J. Math. Anal., 30 (1999), 95-117.
doi: 10.1137/S0036141097322169. |
[17] |
R. Kippenhahn and A. Weigert, Stellar Structure and Evolution Springer Verlag, Berlin-Heidelberg, 1994. Google Scholar |
[18] |
S. Klainerman and A. Majda,
Singular limits of quasilinear hyperbolic systems with large
parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math, 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[19] |
C. Lin,
Asymptotic stability of rarefaction waves in radiation hydrodynamics, Comm. Math. Sci., 9 (2011), 207-223.
doi: 10.4310/CMS.2011.v9.n1.a10. |
[20] |
C. Lin, J. F. Coulombel and Th. Goudon,
Shock profiles for non equilibrium radiating gases, Physica D, 218 (2006), 83-94.
doi: 10.1016/j.physd.2006.04.012. |
[21] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser. A, 55 (1979), 337-342.
doi: 10.3792/pjaa.55.337. |
[22] |
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.
|
[23] |
D. Mihalas and B. Weibel-Mihalas,
Foundations of Radiation Hydrodynamics, Oxford University Press, 1984. |
[24] |
S. S. Penner and D. B. Olfe, Radiation and Reentry, Academic Press, New York, 1968. Google Scholar |
[25] |
G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1973. Google Scholar |
[26] |
C. Rohde and W.-A. Yong,
The nonrelativistic limit in radiation hydrodynamics: I. Weak
entropy solutions for a model problem, J. Diff. Eqns., 234 (2007), 91-109.
doi: 10.1016/j.jde.2006.11.010. |
[27] |
R. N. Thomas, Some Aspects of Non-Equilibrium Thermodynamics in the Presence of a Radiation Field, University of Colorado Press, Boulder, Colorado, 1965. Google Scholar |
[28] |
W. J. Wang and F. Xie,
The initial value problem for a multi-dimensional radiation hydrodynamics model with viscosity,, Math. Methods Appl. Sci., 34 (2011), 776-791.
doi: 10.1002/mma.1398. |
[29] |
Y. B. Zeldovich and Y. P. Raizer, Phsics of Shock Waves and High-Temperture Hydrodynamic Phenomenon, Academic Press, 1966. Google Scholar |
[30] |
X. Zhong and S. Jiang,
Local existence and finite time blow-up in multidimensional radiation
hydrodynamics,, J. Math. Fluid Mech., 9 (2007), 543-564.
doi: 10.1007/s00021-005-0213-3. |
[1] |
Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907 |
[2] |
Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 |
[3] |
Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 |
[4] |
Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845 |
[5] |
Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517 |
[6] |
Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195 |
[7] |
Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143 |
[8] |
Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107 |
[9] |
Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 |
[10] |
Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315 |
[11] |
Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006 |
[12] |
Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 |
[13] |
Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 |
[14] |
Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005 |
[15] |
Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437 |
[16] |
Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015 |
[17] |
Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056 |
[18] |
Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1 |
[19] |
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 & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004 |
[20] |
Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]