# American Institute of Mathematical Sciences

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

 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

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.
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.

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), 385-418. 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.
 [1] 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 [2] Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115 [3] Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332 [4] Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621 [5] Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087 [6] Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333 [7] Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781 [8] Claude-Michael Brauner, Josephus Hulshof, J.-F. Ripoll. Existence of travelling wave solutions in a combustion-radiation model. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 193-208. doi: 10.3934/dcdsb.2001.1.193 [9] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [10] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [11] Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117 [12] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [13] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [14] Binbin Shi, Weike Wang. Existence and blow up of solutions to the $2D$ Burgers equation with supercritical dissipation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1169-1192. doi: 10.3934/dcdsb.2019215 [15] Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051 [16] Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 [17] Mingzhu Wu, Zuodong Yang. Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case. Communications on Pure and Applied Analysis, 2007, 6 (2) : 531-540. doi: 10.3934/cpaa.2007.6.531 [18] Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations and Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019 [19] Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583 [20] Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465

2020 Impact Factor: 1.392