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

Existence results for compressible radiation hydrodynamic equations with vacuum

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 $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.
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

[1]

Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009

[2]

Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure & Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923

[3]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[4]

Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167

[5]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

[6]

Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072

[7]

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

[8]

Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327

[9]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[10]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333

[11]

Xin Zhong. A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3249-3264. doi: 10.3934/dcdsb.2018318

[12]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[13]

Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828

[14]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126

[15]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771

[16]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733

[17]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465

[18]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54

[19]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

[20]

Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]