Articles

INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS

  • Jing ZHANG ,
  • Yongqian ZHANG
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  • School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received date: 2016-12-12

  Revised date: 2017-02-15

  Online published: 2018-02-25

Supported by

This work was supported in part by NSFC Project (11421061), the 111 Project (B08018), and by Shanghai Natural Science Foundation (15ZR1403900).

Abstract

We study the initial-boundary value problem for the one dimensional Euler-Boltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions (Ut,d, It,d) and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.

Cite this article

Jing ZHANG , Yongqian ZHANG . INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS[J]. Acta mathematica scientia, Series B, 2018 , 38(1) : 34 -56 . DOI: 10.1016/S0252-9602(17)30116-9

References

[1] Berthon C, Buet C, et al. Mathematical Models and Numerical Methods for Radiative Transfer. Paris:Société Mathématique de France, 2009
[2] Blanc X, Ducomet B. Global weak solutions to 1D compressible Euler equations with radiation. Commun Math Sci, 2015, 13(7):1905-1936
[3] Bressan A. Hyperbolic Systems of Conservation Laws. Oxford:Oxford University Press, 2000
[4] Conlon J G, Liu T P. Admissible solutions of hyperbolic conservation laws. Mem Amer Math Soc, 1981, 30(240)
[5] Dafermos C M. Hyperbolic Systems of Conservation Laws. Dordrecht:NATO Adv Sci Inst Ser C Math Phys Sci, 1983
[6] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18:697-715
[7] Hong J M, LeFloch P G. A version of the Glimm method based on generalized Riemann problems. Port Math (NS), 2007, 64(2):199-236
[8] Jiang P, Wang D H. Formation of singularities of solutions to the three-dimensional Euler-Boltzmann equations in radiation hydrodynamics. Nonlinearity, 2010, 23(4):809-821
[9] Jiang P, Wang D H. Global weak solutions to the Euler-Boltzmann equations in radiation hydrodynamics. Quart Appl Math, 2012, 70(1):25-44
[10] Jiang P, Wang Y. Existence of solutions to an initial-boundary value problem of multidimensional radiation hydrodynamics. J Differ Equ, 2011, 251(6):1616-1636
[11] Kawashima S, Nikkuni Y, Nishibata S. The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics//Analysis of Systems of Conservation Laws. Chapman & Hall/CRC Monogr Surv Pure Appl Math, 1999
[12] Kawashima S, Tanaka Y. Stability of rarefaction waves for a model system of a radiating gas. Kyushu J Math, 2004, 58(2):211-250
[13] Lin C J. Asymptotic stability of rarefaction waves in radiative hydrodynamics. Ser Contemp Appl Math CAM, 2012, 18:543-550
[14] Liu T P. Quasilinear hyperbolic systems. Comm Math Phys, 1979, 68(2):141-172
[15] Pomraning G C. The Equations of Radiation Hydrodynamics. Pergamon Press, 1973
[16] Rohde C, Xie F. Decay rates to viscous contact waves for a 1D compressible radiation hydrodynamics model. Math Models Methods Appl Sci, 2013, 23(3):441-469
[17] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York:Springer-Verlag, 1994
[18] Tang P F, Fang B X, Wang Y G. On local structural stability of one-dimensional shocks in radiation hydrodynamics. Acta Math Sci, 2015, 35B(1):1-44
[19] Wang Z J. Global existence of a shock front solution to a 1-dimensional piston problem. Chin Ann Math Ser A, 2005, 26(4):549-560
[20] Zhang Y Q. Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary. SIAM J Math Anal, 1999, 31(1):166-183
[21] Zhong X, Jiang S. Local existence and finite-time blow-up in multidimensional radiation hydrodynamics. J Math Fluid Mech, 2007, 9(4):543-564
[22] Zhu Y F, Jiang P. Existence of a shock wave in a one-dimensional radiation hydrodynamic system. Acta Math Sci, 2011, 31A(1):1-17
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