On regular solutions of the 3D compressible isentropic Euler-Boltzmann equations with vacuum

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July 2015, 35(7): 3059-3086. doi: 10.3934/dcds.2015.35.3059

On regular solutions of the $3$D compressible isentropic Euler-Boltzmann equations with vacuum

Yachun Li ^1, and Shengguo Zhu ^2,

1.	Department of Mathematics and Key Lab of Scientific and Engineering Computing (MOE), Shanghai Jiao Tong University, Shanghai 200240
2.	Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received September 2013 Revised November 2014 Published January 2015

Abstract
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In this paper, we discuss the Cauchy problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. We establish the existence of a unique local regular solution with vacuum by the theory of quasi-linear symmetric hyperbolic systems and some techniques dealing with the complexity caused by the coupling between fluid and radiation field under some physical assumptions for the radiation quantities. Moreover, it is interesting to show the non-global existence of regular solutions caused by the effect of vacuum for polytropic gases with adiabatic exponent $1<\gamma\leq 3$ via some observations on the propagation of the radiation field. Compared with [11][15][20], some new initial conditions that will lead to the finite time blow-up for classical solutions have been introduced. These blow-up results tell us that the radiation effect on the fluid is not strong enough to prevent the formation of singularities caused by the appearance of vacuum.

Keywords: local existence, vacuum, blow up., Radiation hydrodynamics, regular solutions.

Mathematics Subject Classification: Primary: 35Q35, 35D35; Secondary: 35A01, 35B4.

Citation: Yachun Li, Shengguo Zhu. On regular solutions of the $3$D compressible isentropic Euler-Boltzmann equations with vacuum. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3059-3086. doi: 10.3934/dcds.2015.35.3059

References:

[1]	C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation hydrodynamics, J. Quant. Spectroscopy Rad. Transf., 85 (2004), 385-418. doi: 10.1016/S0022-4073(03)00233-4.
[2]	B. Ducomet, E. Feireisl and Š. Nečasová, On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré. (C) Non Line. Anal., 28 (2011), 797-812. doi: 10.1016/j.anihpc.2011.06.002.
[3]	B. Ducomet and Š. Nečasová, Global weak solutions to the 1-D compressible Navier-Stokes equations with radiation, Commun. Math. Anal., 8 (2010), 23-65.
[4]	B. Ducomet and Š. Nečasová, Large time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation, Annali di Matematica Pura ed Applicata , 191 (2012), 219-260. doi: 10.1007/s10231-010-0180-z.
[5]	P. Jiang and D. Wang, Formation of singularities of solutions to the three-dimensional Euler-Boltzmann equations in radiation hydrodynamics, Nonlinearity, 23 (2010), 809-821. doi: 10.1088/0951-7715/23/4/003.
[6]	S. Jiang and X. Zhong, Local existence and fiinte-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech., 9 (2007), 543-564. doi: 10.1007/s00021-005-0213-3.
[7]	R. Kippenhahn and A. Weigert, Stellar structure and Evolution, Springer, Berlin, Heidelberg, 1994.
[8]	P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. doi: 10.1063/1.1704154.
[9]	Y. Li and S. Zhu, Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum, J. Differential Equations, 256 (2014), 3943-3980. doi: 10.1016/j.jde.2014.03.007.
[10]	Y. Li and S. Zhu, Existence results for the compressible radiation hydrodynamics equations with vacuum,, 2013, ().
[11]	T. Liu and T. Yang, Compressible Euler equations with vacuum, J. Differential Equations, 140 (1997), 223-237. doi: 10.1006/jdeq.1997.3281.
[12]	T. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32. doi: 10.1007/BF03167296.
[13]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Science, 53, Spinger-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.
[14]	T. Makino, Blowing up solutions of the Euler-Possion equation for the evolution of gaseous stars, Trans. Theo. Statist. Phys., 21 (1992), 615-624. doi: 10.1080/00411459208203801.
[15]	T. Makino, S. Ukai and S. Kawashima, Sur la solution à support compact de equations d'Euler compressible, Japan. J. Appl. Math., 3 (1986), 249-257. doi: 10.1007/BF03167100.
[16]	J. Neumann, Discussion on the Existence and Uniqueness or Multiplicity of Solutions of the Aerodynamical Equations, Collected works of J. Von Neumann, 1949.
[17]	G. Pomrancing, The Equations of Radiation Hydrodynamics, Oxford, Pergamon, 1973.
[18]	T. Sideris, T. Becca and D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with Damping, Commun. Part. Differ. Equations, 28 (2003), 795-816. doi: 10.1081/PDE-120020497.
[19]	T. Sideris, Formation of singulirities in three-dimensional compressible fluids, Commun. Math. Phys., 101 (1985), 475-485. doi: 10.1007/BF01210741.
[20]	Z. Xin and W. Yan, On blow-up of classical solutions to the compressible Navier-Stokes Equations, Commun. Math. Phys., 321 (2013), 529-541. doi: 10.1007/s00220-012-1610-0.
[21]	C. Xu and T. Yang, Local existence with physical vacuum boundary condition to Euler equations with damping, J. Differertial Equations, 210 (2005), 217-231. doi: 10.1016/j.jde.2004.06.005.

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References:

[1]	C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation hydrodynamics, J. Quant. Spectroscopy Rad. Transf., 85 (2004), 385-418. doi: 10.1016/S0022-4073(03)00233-4.
[2]	B. Ducomet, E. Feireisl and Š. Nečasová, On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré. (C) Non Line. Anal., 28 (2011), 797-812. doi: 10.1016/j.anihpc.2011.06.002.
[3]	B. Ducomet and Š. Nečasová, Global weak solutions to the 1-D compressible Navier-Stokes equations with radiation, Commun. Math. Anal., 8 (2010), 23-65.
[4]	B. Ducomet and Š. Nečasová, Large time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation, Annali di Matematica Pura ed Applicata , 191 (2012), 219-260. doi: 10.1007/s10231-010-0180-z.
[5]	P. Jiang and D. Wang, Formation of singularities of solutions to the three-dimensional Euler-Boltzmann equations in radiation hydrodynamics, Nonlinearity, 23 (2010), 809-821. doi: 10.1088/0951-7715/23/4/003.
[6]	S. Jiang and X. Zhong, Local existence and fiinte-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech., 9 (2007), 543-564. doi: 10.1007/s00021-005-0213-3.
[7]	R. Kippenhahn and A. Weigert, Stellar structure and Evolution, Springer, Berlin, Heidelberg, 1994.
[8]	P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. doi: 10.1063/1.1704154.
[9]	Y. Li and S. Zhu, Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum, J. Differential Equations, 256 (2014), 3943-3980. doi: 10.1016/j.jde.2014.03.007.
[10]	Y. Li and S. Zhu, Existence results for the compressible radiation hydrodynamics equations with vacuum,, 2013, ().
[11]	T. Liu and T. Yang, Compressible Euler equations with vacuum, J. Differential Equations, 140 (1997), 223-237. doi: 10.1006/jdeq.1997.3281.
[12]	T. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32. doi: 10.1007/BF03167296.
[13]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Science, 53, Spinger-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.
[14]	T. Makino, Blowing up solutions of the Euler-Possion equation for the evolution of gaseous stars, Trans. Theo. Statist. Phys., 21 (1992), 615-624. doi: 10.1080/00411459208203801.
[15]	T. Makino, S. Ukai and S. Kawashima, Sur la solution à support compact de equations d'Euler compressible, Japan. J. Appl. Math., 3 (1986), 249-257. doi: 10.1007/BF03167100.
[16]	J. Neumann, Discussion on the Existence and Uniqueness or Multiplicity of Solutions of the Aerodynamical Equations, Collected works of J. Von Neumann, 1949.
[17]	G. Pomrancing, The Equations of Radiation Hydrodynamics, Oxford, Pergamon, 1973.
[18]	T. Sideris, T. Becca and D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with Damping, Commun. Part. Differ. Equations, 28 (2003), 795-816. doi: 10.1081/PDE-120020497.
[19]	T. Sideris, Formation of singulirities in three-dimensional compressible fluids, Commun. Math. Phys., 101 (1985), 475-485. doi: 10.1007/BF01210741.
[20]	Z. Xin and W. Yan, On blow-up of classical solutions to the compressible Navier-Stokes Equations, Commun. Math. Phys., 321 (2013), 529-541. doi: 10.1007/s00220-012-1610-0.
[21]	C. Xu and T. Yang, Local existence with physical vacuum boundary condition to Euler equations with damping, J. Differertial Equations, 210 (2005), 217-231. doi: 10.1016/j.jde.2004.06.005.

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