American Institute of Mathematical Sciences

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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

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 & 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.   Google Scholar

[2]

Z. Chen and Y. Wang, The well-posedness of the Cauchy problem for the Navier-Stokes-Boltzmann equations in radiation hydrodynamics,, Dissertation, (2012).   Google Scholar

[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.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities,, Manu. Math., 120 (2006), 91.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[5]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Differential Equations, 228 (2006), 377.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[6]

Y. Cho and B. Jin, Blow-up of viscous heat-conducting compressible flows,, J. Math. Anal. Appl., 320 (2006), 819.  doi: 10.1016/j.jmaa.2005.08.005.  Google Scholar

[7]

H. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Differential Equations, 190 (2003), 504.  doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2011.06.002.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10231-010-0180-z.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1002/cpa.21382.  Google Scholar

[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.  doi: 10.1088/0951-7715/23/4/003.  Google Scholar

[14]

S. Jiang and X. Zhong, Local existence and fiinte-time blow-up in multidimensional radiation hydrodynamics,, J. Math. Fluid Mech., 9 (2007), 543.  doi: 10.1007/s00021-005-0213-3.  Google Scholar

[15]

R. Kippenhahn and A. Weigert, Stellar Structure and Evolution,, Springer, (1994).   Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/j.jde.2014.03.007.  Google Scholar

[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.  doi: 10.1007/BF03167100.  Google Scholar

[19]

G. Pomrancing, The Equations of Radiation Hydrodynamics,, Oxford: Pergamon, (1973).   Google Scholar

[20]

J. Simon, Compact sets in $L^P(0,T;B)$,, Ann. Mat. Pura. Appl, 146 (1986), 65.  doi: 10.1007/BF01762360.  Google Scholar

[21]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, North-Holland: Amsterdam, (1984).   Google Scholar

show all references

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.   Google Scholar

[2]

Z. Chen and Y. Wang, The well-posedness of the Cauchy problem for the Navier-Stokes-Boltzmann equations in radiation hydrodynamics,, Dissertation, (2012).   Google Scholar

[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.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities,, Manu. Math., 120 (2006), 91.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[5]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,, J. Differential Equations, 228 (2006), 377.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[6]

Y. Cho and B. Jin, Blow-up of viscous heat-conducting compressible flows,, J. Math. Anal. Appl., 320 (2006), 819.  doi: 10.1016/j.jmaa.2005.08.005.  Google Scholar

[7]

H. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Differential Equations, 190 (2003), 504.  doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2011.06.002.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10231-010-0180-z.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1002/cpa.21382.  Google Scholar

[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.  doi: 10.1088/0951-7715/23/4/003.  Google Scholar

[14]

S. Jiang and X. Zhong, Local existence and fiinte-time blow-up in multidimensional radiation hydrodynamics,, J. Math. Fluid Mech., 9 (2007), 543.  doi: 10.1007/s00021-005-0213-3.  Google Scholar

[15]

R. Kippenhahn and A. Weigert, Stellar Structure and Evolution,, Springer, (1994).   Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/j.jde.2014.03.007.  Google Scholar

[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.  doi: 10.1007/BF03167100.  Google Scholar

[19]

G. Pomrancing, The Equations of Radiation Hydrodynamics,, Oxford: Pergamon, (1973).   Google Scholar

[20]

J. Simon, Compact sets in $L^P(0,T;B)$,, Ann. Mat. Pura. Appl, 146 (1986), 65.  doi: 10.1007/BF01762360.  Google Scholar

[21]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, North-Holland: Amsterdam, (1984).   Google Scholar

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