Existence results for compressible radiation hydrodynamic equations with vacuum

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May 2015, 14(3): 1023-1052. doi: 10.3934/cpaa.2015.14.1023

Existence results for compressible radiation hydrodynamic 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 2014 Revised January 2015 Published March 2015

Abstract
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In this paper, we consider the three-dimensional compressible isentropic radiation hydrodynamic (RHD) equations. The existence of unique local strong solutions is firstly proved when the initial data are arbitrarily large, contain vacuum and satisfy some initial layer compatibility condition. The initial mass density does not need to be bounded away from zero and may vanish in some open set. We also prove that if the initial vacuum is not so irregular, then the initial layer compatibility condition is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we establish a blow-up criterion for the strong solution that we obtained. The similar results also hold for the barotropic flow with general pressure law $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.

Keywords: Radiation, blow-up criterion., vacuum, Navier-Stokes-Boltzmann equations, strong solutions.

Mathematics Subject Classification: Primary: 35Q35, 35D35; Secondary: 35A01, 35E1.

Citation: Yachun Li, Shengguo Zhu. Existence results for compressible radiation hydrodynamic equations with vacuum. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1023-1052. doi: 10.3934/cpaa.2015.14.1023

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.
[2]	Z. Chen and Y. Wang, The well-posedness of the Cauchy problem for the Navier-Stokes-Boltzmann equations in radiation hydrodynamics, Dissertation, Shanghai Jiao Tong University, 2012.
[3]	Y. Cho, H. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.
[4]	Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manu. Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y.
[5]	Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411. doi: 10.1016/j.jde.2006.05.001.
[6]	Y. Cho and B. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl., 320 (2006), 819-826. doi: 10.1016/j.jmaa.2005.08.005.
[7]	H. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.
[8]	B. Ducomet, E. Feireisl and Š. Nečasová, On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré. (C) Non Line. Anal., 6 (2011), 797-812. doi: 10.1016/j.anihpc.2011.06.002.
[9]	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.
[10]	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, 8 (2012), 219-260. doi: 10.1007/s10231-010-0180-z.
[11]	G. Galdi, An Introduction to the Mathmatical Theorey of the Navier-Stokes Equations, Springer: New York, 1994. doi: 10.1007/978-0-387-09620-9.
[12]	X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.
[13]	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.
[14]	S. Jiang and X. Zhong, Local existence and fiinte-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech., 9 (2007), 543-64. doi: 10.1007/s00021-005-0213-3.
[15]	R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, Springer, Berlin, Heideberg, 1994.
[16]	Y. Li and S. Zhu, On regular solutions of the $3$-D compressible isentropic Euler-Boltzmann equations with vacuum, to appear in Discrete Contin. Dynam. Systems-A, 2015.
[17]	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.
[18]	T. Makino, S. Ukai and S. Kawashima, Sur la solution à support compact de equations d'Euler compressible, Japan. J. Appl. Math., 33 (1986), 249-257. doi: 10.1007/BF03167100.
[19]	G. Pomrancing, The Equations of Radiation Hydrodynamics, Oxford: Pergamon, 1973.
[20]	J. Simon, Compact sets in $L^P(0,T;B)$, Ann. Mat. Pura. Appl, 146 (1986), 65-96. doi: 10.1007/BF01762360.
[21]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland: Amsterdam, 1984.

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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.
[2]	Z. Chen and Y. Wang, The well-posedness of the Cauchy problem for the Navier-Stokes-Boltzmann equations in radiation hydrodynamics, Dissertation, Shanghai Jiao Tong University, 2012.
[3]	Y. Cho, H. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.
[4]	Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manu. Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y.
[5]	Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411. doi: 10.1016/j.jde.2006.05.001.
[6]	Y. Cho and B. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl., 320 (2006), 819-826. doi: 10.1016/j.jmaa.2005.08.005.
[7]	H. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.
[8]	B. Ducomet, E. Feireisl and Š. Nečasová, On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré. (C) Non Line. Anal., 6 (2011), 797-812. doi: 10.1016/j.anihpc.2011.06.002.
[9]	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.
[10]	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, 8 (2012), 219-260. doi: 10.1007/s10231-010-0180-z.
[11]	G. Galdi, An Introduction to the Mathmatical Theorey of the Navier-Stokes Equations, Springer: New York, 1994. doi: 10.1007/978-0-387-09620-9.
[12]	X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.
[13]	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.
[14]	S. Jiang and X. Zhong, Local existence and fiinte-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech., 9 (2007), 543-64. doi: 10.1007/s00021-005-0213-3.
[15]	R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, Springer, Berlin, Heideberg, 1994.
[16]	Y. Li and S. Zhu, On regular solutions of the $3$-D compressible isentropic Euler-Boltzmann equations with vacuum, to appear in Discrete Contin. Dynam. Systems-A, 2015.
[17]	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.
[18]	T. Makino, S. Ukai and S. Kawashima, Sur la solution à support compact de equations d'Euler compressible, Japan. J. Appl. Math., 33 (1986), 249-257. doi: 10.1007/BF03167100.
[19]	G. Pomrancing, The Equations of Radiation Hydrodynamics, Oxford: Pergamon, 1973.
[20]	J. Simon, Compact sets in $L^P(0,T;B)$, Ann. Mat. Pura. Appl, 146 (1986), 65-96. doi: 10.1007/BF01762360.
[21]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland: Amsterdam, 1984.

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