American Institute of Mathematical Sciences

Advanced
July  2019, 18(4): 1567-1599. doi: 10.3934/cpaa.2019075

On the existence of solutions and causality for relativistic viscous conformal fluids

Marcelo M. Disconzi

Department of Mathematics, Vanderbilt University, Nashville, TN 37211, USA

Received  August 2017 Revised  November 2018 Published  January 2019

Fund Project: M. M. D. is partially supported by NSF grant # DMS-1812826, by a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and by a Discovery grant administered by Vanderbilt University.

We consider a stress-energy tensor describing a pure radiation viscous fluid with conformal symmetry introduced in [3]. We show that the corresponding equations of motions are causal in Minkowski background and also when coupled to Einstein's equations, and solve the associated initial-value problem.

Keywords: Relativistic viscous fluids, conformal fluids, Einstein's equations, causality, well-posedness.
Mathematics Subject Classification: Primary: 35Q76; Secondary: 35L15.
Citation: Marcelo M. Disconzi. On the existence of solutions and causality for relativistic viscous conformal fluids. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1567-1599. doi: 10.3934/cpaa.2019075
References:
[1]

A. M. Anile, Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1 edition, 1990.

[2]

R. Baier, P. Romatschke, D. T. Son, A. O. Starinets and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, JHEP, 04 (2008), 100. doi: 10.1088/1126-6708/2008/04/100.

[3]

F. Bemfica, M. M. Disconzi and J. Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Physical Review D, 98 (2018), 104064 (26 pages).

[4]

S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP, 02 (2008), 045.

[5]

C. H. Chan, M. Czubak and M. M. Disconzi, The formulation of the Navier-Stokes equations on Riemannian manifolds, Journal of Geometry and Physics, 121 (2017), 335-346.  doi: 10.1016/j.geomphys.2017.07.015.

[6]

Y. Choquet-Bruhat, Diagonalisation des systèmes quasi-linéaires et hyperbolicité non stricte, J. Math. Pures Appl. (9), 45 (1966), 371-386.

[7] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, New York, 2009. 
[8]

P. T. Chruściel and E. Delay, Manifold structures for sets of solutions of the general relativistic constraint equations, J. Geom. Phys., 51 (2004), 442-472.  doi: 10.1016/j.geomphys.2003.12.002.

[9]

C. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 2, 1st edition, John Wiley & Sons, Inc., 1991.

[10]

M. Czubak and M. M. Disconzi, On the well-posedness of relativistic viscous fluids with non-zero vorticity, Journal of Mathematical Physics, 57 (2016), 042501, 21 pages. doi: 10.1063/1.4944910.

[11]

R. D. de Souza, T. Koide and T. Kodama, Hydrodynamic approaches in relativistic heavy ion reactions, Prog. Part. Nucl. Phys., 86 (2016), 35-85. 

[12]

M. M. Disconzi, On the well-posedness of relativistic viscous fluids, Nonlinearity, 27 (2014), 1915-1935.  doi: 10.1088/0951-7715/27/8/1915.

[13]

M. M. Disconzi, Remarks on the Einstein-Euler-entropy system, Reviews in Mathematical Physics, 27 (2015), 1550014, 45 pages. doi: 10.1142/S0129055X15500142.

[14]

M. M. Disconzi and D. G. Ebin, The free boundary Euler equations with large surface tension, Journal of Differential Equations, 261 (2016), 821-889.  doi: 10.1016/j.jde.2016.03.029.

[15]

M. M. Disconzi, T. W. Kephart and R. J. Scherrer, A new approach to cosmological bulk viscosity, Physical Review D, 91 (2015), 043532 (6 pages). doi: 10.1103/PhysRevD.91.043532.

[16]

M. M. Disconzi, T. W. Kephart and R. J. Scherrer, On a viable first order formulation of relativistic viscous fluids and its applications to cosmology, International Journal of Modern Physics D, 26 (2017), 1750146 (52 pages). doi: 10.1142/S0218271817501462.

[17]

M. M. Discozni and J. Speck, The relativistic euler equations: Remarkable null structures and regularity properties, arXiv: 1809.06204.

[18]

M. Hadžić, S. Shkoller and J. Speck, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, arXiv: 1511.07467.

[19]

G. S. Hall, Weyl manifolds and connections, Journal of Mathematical Physics, 33 (1992), 2633-2638.  doi: 10.1063/1.529582.

[20]

Y. Hatta, J. Noronha and B.-W. Xiao, Exact analytical solutions of second-order conformal hydrodynamics, Physical Review D, 89 (2014), 051702.

[21]

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1975.

[22]

W. A. Hiscock and L. Lindblom, Stability and causality in dissipative relativistic fluids, Annals of Physics, 151 (1983), 466-496.  doi: 10.1016/0003-4916(83)90288-9.

[23]

W. A. Hiscock and L. Lindblom, Generic instabilities in first-order dissipative fluid theories, Phys. Rev. D, 31 (1985), 725-733.  doi: 10.1103/PhysRevD.31.725.

[24]

J. Jang, P. G. LeFloch and N. Masmoudi, Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum, Journal of Differential Equations, 260 (2016), 5481-5509.  doi: 10.1016/j.jde.2015.12.004.

[25]

S. Klainerman and F. Nicolo, The Evolution Problem in General Relativity, Progress in Mathematical Physics, vol. 25, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-2084-8.

[26]

J. Leray, Hyperbolic Differential Equations, The Institute for Advanced Study, Princeton, N. J., 1953.

[27]

J. Leray and Y. Ohya, Systèmes linéaires, hyperboliques non stricts, in Deuxième Colloq. l'Anal. Fonct, Centre Belge Recherches Math., Librairie Universitaire, Louvain, 1964, 105-144.

[28]

J. Leray and Y. Ohya, Équations et systèmes non-linéaires, hyperboliques nonstricts, Math. Ann., 170 (1967), 167-205.  doi: 10.1007/BF01350150.

[29]

J. Leray and Y. Ohya, équations et systèmes non linéaires, hyperboliques non-stricts, in Hyperbolic Equations and Waves Rencontres, Battelle Res. Inst., Seattle, Wash., 1968, Springer, Berlin, 1970, 331-369.

[30]

A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics: Lectures on the Existence of Solutions, W. A. Benjamin, New York, 1967.

[31]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392.  doi: 10.1007/s00220-005-1406-6.

[32]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 3, Dunod, Paris, 1970, Travaux et Recherches Math´ematiques, No. 20.

[33]

R. Loganayagam, Entropy current in conformal hydrodynamics, JHEP, 05 (2008), 087. doi: 10.1088/1126-6708/2008/05/087.

[34] S. Mizohata, On the Cauchy Problem, Science Press and Academic Press, Inc., Hong Kong, 1985. 
[35]

G. Pichon, Étude relativiste de fluides visqueux et chargés, Annales de l'I.H.P. Physique théorique, 2 (1965), 21–85.

[36]

A. D. Rendall, The initial value problem for a class of general relativistic fluid bodies, J. Math. Phys., 33 (1992), 1047-1053.  doi: 10.1063/1.529766.

[37] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, New York, 2013. 
[38]

H. Ringstrom, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2009. doi: 10.4171/053.

[39]

L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, Singapore, 1993. doi: 10.1142/9789814360036.

[40]

R. M. Wald, General Relativity, University of Chicago press, 2010. doi: 10.7208/chicago/9780226870373.001.0001.

show all references

References:
[1]

A. M. Anile, Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1 edition, 1990.

[2]

R. Baier, P. Romatschke, D. T. Son, A. O. Starinets and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, JHEP, 04 (2008), 100. doi: 10.1088/1126-6708/2008/04/100.

[3]

F. Bemfica, M. M. Disconzi and J. Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Physical Review D, 98 (2018), 104064 (26 pages).

[4]

S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP, 02 (2008), 045.

[5]

C. H. Chan, M. Czubak and M. M. Disconzi, The formulation of the Navier-Stokes equations on Riemannian manifolds, Journal of Geometry and Physics, 121 (2017), 335-346.  doi: 10.1016/j.geomphys.2017.07.015.

[6]

Y. Choquet-Bruhat, Diagonalisation des systèmes quasi-linéaires et hyperbolicité non stricte, J. Math. Pures Appl. (9), 45 (1966), 371-386.

[7] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, New York, 2009. 
[8]

P. T. Chruściel and E. Delay, Manifold structures for sets of solutions of the general relativistic constraint equations, J. Geom. Phys., 51 (2004), 442-472.  doi: 10.1016/j.geomphys.2003.12.002.

[9]

C. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 2, 1st edition, John Wiley & Sons, Inc., 1991.

[10]

M. Czubak and M. M. Disconzi, On the well-posedness of relativistic viscous fluids with non-zero vorticity, Journal of Mathematical Physics, 57 (2016), 042501, 21 pages. doi: 10.1063/1.4944910.

[11]

R. D. de Souza, T. Koide and T. Kodama, Hydrodynamic approaches in relativistic heavy ion reactions, Prog. Part. Nucl. Phys., 86 (2016), 35-85. 

[12]

M. M. Disconzi, On the well-posedness of relativistic viscous fluids, Nonlinearity, 27 (2014), 1915-1935.  doi: 10.1088/0951-7715/27/8/1915.

[13]

M. M. Disconzi, Remarks on the Einstein-Euler-entropy system, Reviews in Mathematical Physics, 27 (2015), 1550014, 45 pages. doi: 10.1142/S0129055X15500142.

[14]

M. M. Disconzi and D. G. Ebin, The free boundary Euler equations with large surface tension, Journal of Differential Equations, 261 (2016), 821-889.  doi: 10.1016/j.jde.2016.03.029.

[15]

M. M. Disconzi, T. W. Kephart and R. J. Scherrer, A new approach to cosmological bulk viscosity, Physical Review D, 91 (2015), 043532 (6 pages). doi: 10.1103/PhysRevD.91.043532.

[16]

M. M. Disconzi, T. W. Kephart and R. J. Scherrer, On a viable first order formulation of relativistic viscous fluids and its applications to cosmology, International Journal of Modern Physics D, 26 (2017), 1750146 (52 pages). doi: 10.1142/S0218271817501462.

[17]

M. M. Discozni and J. Speck, The relativistic euler equations: Remarkable null structures and regularity properties, arXiv: 1809.06204.

[18]

M. Hadžić, S. Shkoller and J. Speck, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, arXiv: 1511.07467.

[19]

G. S. Hall, Weyl manifolds and connections, Journal of Mathematical Physics, 33 (1992), 2633-2638.  doi: 10.1063/1.529582.

[20]

Y. Hatta, J. Noronha and B.-W. Xiao, Exact analytical solutions of second-order conformal hydrodynamics, Physical Review D, 89 (2014), 051702.

[21]

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1975.

[22]

W. A. Hiscock and L. Lindblom, Stability and causality in dissipative relativistic fluids, Annals of Physics, 151 (1983), 466-496.  doi: 10.1016/0003-4916(83)90288-9.

[23]

W. A. Hiscock and L. Lindblom, Generic instabilities in first-order dissipative fluid theories, Phys. Rev. D, 31 (1985), 725-733.  doi: 10.1103/PhysRevD.31.725.

[24]

J. Jang, P. G. LeFloch and N. Masmoudi, Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum, Journal of Differential Equations, 260 (2016), 5481-5509.  doi: 10.1016/j.jde.2015.12.004.

[25]

S. Klainerman and F. Nicolo, The Evolution Problem in General Relativity, Progress in Mathematical Physics, vol. 25, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-2084-8.

[26]

J. Leray, Hyperbolic Differential Equations, The Institute for Advanced Study, Princeton, N. J., 1953.

[27]

J. Leray and Y. Ohya, Systèmes linéaires, hyperboliques non stricts, in Deuxième Colloq. l'Anal. Fonct, Centre Belge Recherches Math., Librairie Universitaire, Louvain, 1964, 105-144.

[28]

J. Leray and Y. Ohya, Équations et systèmes non-linéaires, hyperboliques nonstricts, Math. Ann., 170 (1967), 167-205.  doi: 10.1007/BF01350150.

[29]

J. Leray and Y. Ohya, équations et systèmes non linéaires, hyperboliques non-stricts, in Hyperbolic Equations and Waves Rencontres, Battelle Res. Inst., Seattle, Wash., 1968, Springer, Berlin, 1970, 331-369.

[30]

A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics: Lectures on the Existence of Solutions, W. A. Benjamin, New York, 1967.

[31]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392.  doi: 10.1007/s00220-005-1406-6.

[32]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 3, Dunod, Paris, 1970, Travaux et Recherches Math´ematiques, No. 20.

[33]

R. Loganayagam, Entropy current in conformal hydrodynamics, JHEP, 05 (2008), 087. doi: 10.1088/1126-6708/2008/05/087.

[34] S. Mizohata, On the Cauchy Problem, Science Press and Academic Press, Inc., Hong Kong, 1985. 
[35]

G. Pichon, Étude relativiste de fluides visqueux et chargés, Annales de l'I.H.P. Physique théorique, 2 (1965), 21–85.

[36]

A. D. Rendall, The initial value problem for a class of general relativistic fluid bodies, J. Math. Phys., 33 (1992), 1047-1053.  doi: 10.1063/1.529766.

[37] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, New York, 2013. 
[38]

H. Ringstrom, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2009. doi: 10.4171/053.

[39]

L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, Singapore, 1993. doi: 10.1142/9789814360036.

[40]

R. M. Wald, General Relativity, University of Chicago press, 2010. doi: 10.7208/chicago/9780226870373.001.0001.

[1]

Juliana Honda Lopes, Gabriela Planas. Well-posedness for a non-isothermal flow of two viscous incompressible fluids. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2455-2477. doi: 10.3934/cpaa.2018117
[2]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[3]

Fabio S. Bemfica, Marcelo M. Disconzi, P. Jameson Graber. Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2885-2914. doi: 10.3934/cpaa.2021068
[4]

Zhenduo Fan, Wenjun Liu, Shengqian Chen. Global well-posedness of the three-dimensional viscous primitive equations with bounded delays. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022019
[5]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[6]

Eduard Feireisl. Mathematical theory of viscous fluids: Retrospective and future perspectives. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 533-555. doi: 10.3934/dcds.2010.27.533
[7]

Eduard Feireisl, Antonín Novotný. Two phase flows of compressible viscous fluids. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2215-2232. doi: 10.3934/dcdss.2022091
[8]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations and Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[9]

Eugenio Aulisa, Lidia Bloshanskaya, Akif Ibragimov. Well productivity index for compressible fluids and gases. Evolution Equations and Control Theory, 2016, 5 (1) : 1-36. doi: 10.3934/eect.2016.5.1
[10]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations and Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007
[11]

Adalet Hanachi, Haroune Houamed, Mohamed Zerguine. On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6473-6506. doi: 10.3934/dcds.2020287
[12]

Diego Rapoport. Random representations of viscous fluids and the passive magnetic fields transported on them. Conference Publications, 2001, 2001 (Special) : 327-336. doi: 10.3934/proc.2001.2001.327
[13]

Allen Montz, Hamid Bellout, Frederick Bloom. Existence and uniqueness of steady flows of nonlinear bipolar viscous fluids in a cylinder. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2107-2128. doi: 10.3934/dcdsb.2015.20.2107
[14]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden, Adele Manes. Energy stability for thermo-viscous fluids with a fading memory heat flux. Evolution Equations and Control Theory, 2015, 4 (3) : 265-279. doi: 10.3934/eect.2015.4.265
[15]

Pitágoras Pinheiro de Carvalho, Juan Límaco, Denilson Menezes, Yuri Thamsten. Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021043
[16]

Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087
[17]

Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255
[18]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361
[19]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813
[20]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure and Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (250)
  • HTML views (207)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy