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

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

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.

Citation: Marcelo M. Disconzi. On the existence of solutions and causality for relativistic viscous conformal fluids. Communications on Pure & 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. ChanM. 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 SouzaT. 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. JangP. 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. ChanM. 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 SouzaT. 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. JangP. 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.

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