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On the existence of solutions and causality for relativistic viscous conformal fluids
Department of Mathematics, Vanderbilt University, Nashville, TN 37211, USA |
We consider a stress-energy tensor describing a pure radiation viscous fluid with conformal symmetry introduced in [
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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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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