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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 & 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. Google Scholar

[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.  Google Scholar

[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). Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

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

[9]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

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

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

[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.  Google Scholar

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

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

[39]

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

[40]

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

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. Google Scholar

[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.  Google Scholar

[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). Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

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

[9]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

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

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

[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.  Google Scholar

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

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

[39]

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

[40]

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

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