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. 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. 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.  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 SouzaT. 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. 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.  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. 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.  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 SouzaT. 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. 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.  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

[1]

Juliana Honda Lopes, Gabriela Planas. Well-posedness for a non-isothermal flow of two viscous incompressible fluids. Communications on Pure & 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 & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253

[11]

Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657

[12]

Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455

[13]

Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517

[14]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395

[15]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

[16]

Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903

[17]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143

[18]

Can Li, Weihua Deng, Lijing Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1989-2015. doi: 10.3934/dcdsb.2019026

[19]

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

[20]

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

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (100)
  • HTML views (165)
  • Cited by (0)

Other articles
by authors

[Back to Top]