July 2018, 38(7): 3239-3268. doi: 10.3934/dcds.2018141

Low Mach number limit for a model of accretion disk

1. 

Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, 67100 L'Aquila, Italy

2. 

Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, 61 avenue du Général de Gaulle, 94010 Créteil Cedex 10, France

3. 

Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic

Received  April 2017 Revised  January 2018 Published  April 2018

We study an hydrodynamical model describing the motion of thick astrophysical disks relying on compressible Navier-Stokes-Fourier-Poisson system. We also suppose that the medium is electrically charged and we include energy exchanges through radiative transfer. Supposing that the system is rotating, we study the singular limit of the system when the Mach number, the Alfven number and Froude number go to zero and we prove convergence to a 3D incompressible MHD system with radiation with two stationary linear transport equations for transport of radiation intensity.

Citation: Donatella Donatelli, Bernard Ducomet, Šárka Nečasová. Low Mach number limit for a model of accretion disk. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3239-3268. doi: 10.3934/dcds.2018141
References:
[1]

R. Balian, From Microphysics to Macrophysics. Methods and Applications of Statistical Physics Vol. II, Springer Verlag, Berlin, Heidelberg, New York, 1992. doi: 10.1007/978-3-540-45475-5.

[2]

C. BardosF. GolseB. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1.

[3]

A. R. Choudhuri, The Physics of Fluids and Plasmas, an Introduction for Astrophysicists, Cambridge University Press, 1998.

[4]

L. DieningM. Ružička and K. Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math., 35 (2010), 87-114. doi: 10.5186/aasfm.2010.3506.

[5]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[6]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 266 (2006), 595-629. doi: 10.1007/s00220-006-0052-y.

[7]

B. DucometE. Feireisl and Š. Nečasová, On a model of radiation hydrodynamics, Ann. I. H. Poincaré-AN, 28 (2011), 797-812. doi: 10.1016/j.anihpc.2011.06.002.

[8]

B. DucometE. FeireislH. Petzeltová and I. Straškraba, Global in time weak solutions for compressible barotropic self-gravitating fluids, Discrete and Continuous Dynamical Systems, 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113.

[9]

B. DucometM. Kobera and Š. Nečasová, Global existence of a weak solution for a model in radiation hydrodynamics, Acta Applicandae Mathematicae, 150 (2017), 43-65. doi: 10.1007/s10440-016-0093-y.

[10]

B. Ducomet, M. Caggio, Š. Nečasová and M. Pokorný, The rotating Navier-Stokes-Fourier-Poisson system on thin domains, Submitted.

[11]

B. DucometŠ. NečasováM. Pokorný and M. A. Rodriguez-Bellido, Derivation of the Navier-Stokes-Poisson system for an accretion disk, Accepted in JMFM, (2018), 1-23. doi: 10.1007/s00021-017-0358-x.

[12]

B. Ducomet and Š. Nečasová, Low Mach number limit for a model of radiative flow, J. Evol. Equ., 14 (2014), 357-385. doi: 10.1007/s00028-014-0217-7.

[13]

G. Duvaut and J. -L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Heidelberg, 1976.

[14]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhauser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[15]

X Hu and D. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294. doi: 10.1137/080723983.

[16]

S. JiangQ. JuF. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420. doi: 10.1016/j.aim.2014.03.022.

[17]

S. JiangQ. Ju and F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365. doi: 10.1088/0951-7715/25/5/1351.

[18]

S. JiangQ. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal., 42 (2010), 2539-2553. doi: 10.1137/100785168.

[19]

P. Kukučka, Singular limits of the equations of magnetohydrodynamics, J. Math. Fluid Mech., 13 (2011), 173-189. doi: 10.1007/s00021-009-0007-0.

[20]

P. Kukučka, On the existence of finite energy weak solutions to the Navier-Stokes equations in irregular domains, Math. Methods Appl. Sci., 32 (2009), 1428-1451. doi: 10.1002/mma.1101.

[21]

Y. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations, 251 (2011), 1990-2023. doi: 10.1016/j.jde.2011.04.016.

[22]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, AMS 1968.

[23]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, A. M. S., 1997.

[24]

Matias Montesinos Armijo, Review: Accretion disk theory, arXiv: 1203.6851v1, [astro-ph.HE] 30 Mar 2012.

[25]

J. Nečas and T. Roubíček, Buoyancy-driven viscous flow with $ L^1$-data, Nonlinear Analysis, 46 (2001), 737-755. doi: 10.1016/S0362-546X(01)00676-9.

[26]

A. Novotný and I. Straškraba. Introduction to the Mathematical Theory of Compressible Flows, Oxford Lectures Series in Mathematics and Its Applications, Vol. 27, Oxford University Press, 2004.

[27]

A. NovotnýM. Růžička and G. Thäter, Singular limit of the equations of magnetohydrodynamics in the presence of strong stratification, Math. Models Methods Appl. Sci., 21 (2011), 115-147. doi: 10.1142/S0218202511005003.

[28]

G. I. Ogilvie, Accretion disks, in "Fluid Dynamics and Dynamos in Astrophysics and Geophysics", A. M. Soward, C. A. Jones, D. W. Hughes, N. O. Weiss Edts., CRC Press, 12(2005), 1-28.

[29]

T. Padmanabhan, Theoretical Astrophysics, Vol Ⅱ: Stars and Stellar Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511840159.

[30]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Bikhäuser, 2005.

[31]

M. Sermange and R. Temam, Some mathematical questions related to magnetohydrodynamics, Comm. Pure Appl. Math., 36 (1983), 635-664.

[32]

R. Temam, Navier-Stokes Equations, Amsterdam-New York, 1979.

show all references

References:
[1]

R. Balian, From Microphysics to Macrophysics. Methods and Applications of Statistical Physics Vol. II, Springer Verlag, Berlin, Heidelberg, New York, 1992. doi: 10.1007/978-3-540-45475-5.

[2]

C. BardosF. GolseB. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1.

[3]

A. R. Choudhuri, The Physics of Fluids and Plasmas, an Introduction for Astrophysicists, Cambridge University Press, 1998.

[4]

L. DieningM. Ružička and K. Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math., 35 (2010), 87-114. doi: 10.5186/aasfm.2010.3506.

[5]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[6]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 266 (2006), 595-629. doi: 10.1007/s00220-006-0052-y.

[7]

B. DucometE. Feireisl and Š. Nečasová, On a model of radiation hydrodynamics, Ann. I. H. Poincaré-AN, 28 (2011), 797-812. doi: 10.1016/j.anihpc.2011.06.002.

[8]

B. DucometE. FeireislH. Petzeltová and I. Straškraba, Global in time weak solutions for compressible barotropic self-gravitating fluids, Discrete and Continuous Dynamical Systems, 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113.

[9]

B. DucometM. Kobera and Š. Nečasová, Global existence of a weak solution for a model in radiation hydrodynamics, Acta Applicandae Mathematicae, 150 (2017), 43-65. doi: 10.1007/s10440-016-0093-y.

[10]

B. Ducomet, M. Caggio, Š. Nečasová and M. Pokorný, The rotating Navier-Stokes-Fourier-Poisson system on thin domains, Submitted.

[11]

B. DucometŠ. NečasováM. Pokorný and M. A. Rodriguez-Bellido, Derivation of the Navier-Stokes-Poisson system for an accretion disk, Accepted in JMFM, (2018), 1-23. doi: 10.1007/s00021-017-0358-x.

[12]

B. Ducomet and Š. Nečasová, Low Mach number limit for a model of radiative flow, J. Evol. Equ., 14 (2014), 357-385. doi: 10.1007/s00028-014-0217-7.

[13]

G. Duvaut and J. -L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Heidelberg, 1976.

[14]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhauser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[15]

X Hu and D. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294. doi: 10.1137/080723983.

[16]

S. JiangQ. JuF. Li and Z. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420. doi: 10.1016/j.aim.2014.03.022.

[17]

S. JiangQ. Ju and F. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365. doi: 10.1088/0951-7715/25/5/1351.

[18]

S. JiangQ. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal., 42 (2010), 2539-2553. doi: 10.1137/100785168.

[19]

P. Kukučka, Singular limits of the equations of magnetohydrodynamics, J. Math. Fluid Mech., 13 (2011), 173-189. doi: 10.1007/s00021-009-0007-0.

[20]

P. Kukučka, On the existence of finite energy weak solutions to the Navier-Stokes equations in irregular domains, Math. Methods Appl. Sci., 32 (2009), 1428-1451. doi: 10.1002/mma.1101.

[21]

Y. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations, 251 (2011), 1990-2023. doi: 10.1016/j.jde.2011.04.016.

[22]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, AMS 1968.

[23]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, A. M. S., 1997.

[24]

Matias Montesinos Armijo, Review: Accretion disk theory, arXiv: 1203.6851v1, [astro-ph.HE] 30 Mar 2012.

[25]

J. Nečas and T. Roubíček, Buoyancy-driven viscous flow with $ L^1$-data, Nonlinear Analysis, 46 (2001), 737-755. doi: 10.1016/S0362-546X(01)00676-9.

[26]

A. Novotný and I. Straškraba. Introduction to the Mathematical Theory of Compressible Flows, Oxford Lectures Series in Mathematics and Its Applications, Vol. 27, Oxford University Press, 2004.

[27]

A. NovotnýM. Růžička and G. Thäter, Singular limit of the equations of magnetohydrodynamics in the presence of strong stratification, Math. Models Methods Appl. Sci., 21 (2011), 115-147. doi: 10.1142/S0218202511005003.

[28]

G. I. Ogilvie, Accretion disks, in "Fluid Dynamics and Dynamos in Astrophysics and Geophysics", A. M. Soward, C. A. Jones, D. W. Hughes, N. O. Weiss Edts., CRC Press, 12(2005), 1-28.

[29]

T. Padmanabhan, Theoretical Astrophysics, Vol Ⅱ: Stars and Stellar Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511840159.

[30]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Bikhäuser, 2005.

[31]

M. Sermange and R. Temam, Some mathematical questions related to magnetohydrodynamics, Comm. Pure Appl. Math., 36 (1983), 635-664.

[32]

R. Temam, Navier-Stokes Equations, Amsterdam-New York, 1979.

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