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June  2021, 20(6): 2279-2290. doi: 10.3934/cpaa.2021069

Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics

Fabio Sperotto Bemfica 1, , Marcelo Mendes Disconzi 2,, , Casey Rodriguez 3, and  Yuanzhen Shao 4,

1. 

Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, Natal, RN 59072-970, Brazil
2. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
3. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
4. 

Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA

* Corresponding author

Received  August 2020 Revised  February 2021 Published  June 2021 Early access  April 2021

Fund Project: The first author is supported by a Discovery grant administered by Vanderbilt University The second author is supported by a Sloan Research Fellowship, NSF grant # 1812826, a Dean's Faculty Fellowship and a Discovery grant administered by Vanderbilt University. The third author is supported by NSF Postdoctoral Research Fellowship DMS-1703180

In this manuscript, we study the theory of conformal relativistic viscous hydrodynamics introduced in [4], which provided a causal and stable first-order theory of relativistic fluids with viscosity. Local existence and uniqueness of solutions to its equations of motion have been previously established in Gevrey spaces. Here, we improve this result by proving local existence and uniqueness of solutions in Sobolev spaces.

Keywords: Relativistic viscous fluids, conformal symmetry, causality, local existence and uniqueness.
Mathematics Subject Classification: Primary: 35Q75; Secondary: 35Q35, 35Q31.
Citation: Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2279-2290. doi: 10.3934/cpaa.2021069
References:
[1]

M. G. Alford, L. Bovard, M. Hanauske, L. Rezzolla and K. Schwenzer, Viscous dissipation and heat conduction in binary neutron-star mergers, Phys. Rev. Lett., 120 (2018), 041101.

[2]

A. M. Anile, Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics (cambridge monographs on mathematical physics), Cambridge University Press; 1$^st$ edition, 1990.

[3]

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.

[4]

F. S. Bemfica, M. M. Disconzi and J. Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Phys. Rev. D 98 (2018), 104064. doi: 10.1103/physrevd. 98.104064.

[5]

F. S. Bemfica, M. M. Disconzi and J. Noronha, Causality of the Einstein-Israel-Stewart theory with bulk viscosity, Phys. Rev. Lett., 122 (2019), 221602.

[6]

F. S. Bemfica, M. M. Disconzi and J. Noronha, Nonlinear causality of general first-order relativistic viscous hydrodynamics, Physical Review D, 100 (2019), 104020. doi: 10.1103/physrevd. 100.104020.

[7]

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

[8]

U. Brauer and L. Karp, Well-posedness of the Einstein-Euler system in asymptotically flat space-times: the constraint equations, J. Differ. Equ., 251 (2011), 1428-1446.  doi: 10.1016/j.jde.2011.05.037.

[9]

U. Brauer and L. Karp, Local existence of solutions of self gravitating relativistic perfect fluids, Commun. Math. Phys., 325 (2014), 105-141.  doi: 10.1007/s00220-013-1854-3.

[10] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, New York, 2009. 
[11]

D. Christodoulou, The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/031.

[12]

D. Christodoulou, The Shock Development Problem, EMS Monographs in Mathematics, European Mathematical Society (EMS), 2019. doi: 10.4171/192.

[13]

M. Czubak and M. M. Disconzi, On the well-posedness of relativistic viscous fluids with non-zero vorticity, J. Math. Phys., 57 (2016), 042501. doi: 10.1063/1.4944910.

[14]

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

[15]

M. M. Disconzi, Remarks on the Einstein-Euler-entropy system, Rev. Math. Phys., 27 (2015), 1550014. doi: 10.1142/S0129055X15500142.

[16]

M. M. Disconzi, On the existence of solutions and causality for relativistic viscous conformal fluids, Commun. Pure Appl. Anal., 18 (2019), 1567-1599.  doi: 10.3934/cpaa.2019075.

[17]

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

[18]

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. Gravitation, Astrophysics, Cosmology, 26 (2017), 1750146. doi: 10.1142/S0218271817501462.

[19]

Y. Fourès-Bruhat, Théorèmes d'existence en mécanique des fluides relativistes, Bull. Soc. Math. France, 86 (1958), 155-175. 

[20]

G. Fournodavlos and V. Schlue, On "hard stars" in general relativity, Annales Henri Poincaré, 20 (2019), 2135-2172.  doi: 10.1007/s00023-019-00793-4.

[21]

R. Geroch and L. Lindblom, Causal theories of dissipative relativistic fluids, Ann. Physics, 207 (1991), 394-416.  doi: 10.1016/0003-4916(91)90063-E.

[22]

D. Ginsberg, A priori estimates for a relativistic liquid with free surface boundary, J. Hyperbolic Differ. Equ., 16 (2019), 401-442.  doi: 10.1142/S0219891619500152.

[23]

M. Hadžić, S. Shkoller and J. Speck, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, Commun. Partial Differ. Equ., 44 (2019), 859-906.  doi: 10.1080/03605302.2019.1583250.

[24]

U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Ann. Rev. Nucl. Part. Sci., 63 (2013), 123-151. 

[25]

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

[26]

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.

[27]

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

[28]

L. Lehner, O. A. Reula and M. E. Rubio, Hyperbolic theory of relativistic conformal dissipative fluids, Phys. Rev. D, 97 (2018), 024013. doi: 10.1103/physrevd. 97.024013.

[29]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. III, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 183.

[30]

T. A. Oliynyk, Dynamical relativistic liquid bodies I: constraint propagation, preprint, arXiv: 1707.08219.

[31]

T. A. Oliynyk, A priori estimates for relativistic liquid bodies, Bull. Sci. Math., 141 (2017), 105-222.  doi: 10.1016/j.bulsci.2017.02.001.

[32]

T. A. Oliynyk, Dynamical relativistic liquid bodies, preprint, arXiv: 1907.08192.

[33] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, New York, 2013. 
[34]

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

[35]

M. Strickland, Anisotropic hydrodynamics: Motivation and methodology, Nucl. Phys. A, 926 (2014), 92-101.  doi: 10.5506/APhysPolB.45.2355.

[36]

M. Strickland, Anisotropic hydrodynamics: Three lectures, Acta Phys. Polon. B, 45 (2014), 2355-2394.  doi: 10.5506/APhysPolB.45.2355.

[37]

M. E. Taylor, Partial Differential Equations III: Nonlinear Equation, Spring, New York, 2010. doi: 10.1007/978-1-4419-7049-7.

[38] S. Weinberg, Cosmology, Oxford University Press, 2008. 

show all references

References:
[1]

M. G. Alford, L. Bovard, M. Hanauske, L. Rezzolla and K. Schwenzer, Viscous dissipation and heat conduction in binary neutron-star mergers, Phys. Rev. Lett., 120 (2018), 041101.

[2]

A. M. Anile, Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics (cambridge monographs on mathematical physics), Cambridge University Press; 1$^st$ edition, 1990.

[3]

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.

[4]

F. S. Bemfica, M. M. Disconzi and J. Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Phys. Rev. D 98 (2018), 104064. doi: 10.1103/physrevd. 98.104064.

[5]

F. S. Bemfica, M. M. Disconzi and J. Noronha, Causality of the Einstein-Israel-Stewart theory with bulk viscosity, Phys. Rev. Lett., 122 (2019), 221602.

[6]

F. S. Bemfica, M. M. Disconzi and J. Noronha, Nonlinear causality of general first-order relativistic viscous hydrodynamics, Physical Review D, 100 (2019), 104020. doi: 10.1103/physrevd. 100.104020.

[7]

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

[8]

U. Brauer and L. Karp, Well-posedness of the Einstein-Euler system in asymptotically flat space-times: the constraint equations, J. Differ. Equ., 251 (2011), 1428-1446.  doi: 10.1016/j.jde.2011.05.037.

[9]

U. Brauer and L. Karp, Local existence of solutions of self gravitating relativistic perfect fluids, Commun. Math. Phys., 325 (2014), 105-141.  doi: 10.1007/s00220-013-1854-3.

[10] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, New York, 2009. 
[11]

D. Christodoulou, The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/031.

[12]

D. Christodoulou, The Shock Development Problem, EMS Monographs in Mathematics, European Mathematical Society (EMS), 2019. doi: 10.4171/192.

[13]

M. Czubak and M. M. Disconzi, On the well-posedness of relativistic viscous fluids with non-zero vorticity, J. Math. Phys., 57 (2016), 042501. doi: 10.1063/1.4944910.

[14]

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

[15]

M. M. Disconzi, Remarks on the Einstein-Euler-entropy system, Rev. Math. Phys., 27 (2015), 1550014. doi: 10.1142/S0129055X15500142.

[16]

M. M. Disconzi, On the existence of solutions and causality for relativistic viscous conformal fluids, Commun. Pure Appl. Anal., 18 (2019), 1567-1599.  doi: 10.3934/cpaa.2019075.

[17]

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

[18]

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. Gravitation, Astrophysics, Cosmology, 26 (2017), 1750146. doi: 10.1142/S0218271817501462.

[19]

Y. Fourès-Bruhat, Théorèmes d'existence en mécanique des fluides relativistes, Bull. Soc. Math. France, 86 (1958), 155-175. 

[20]

G. Fournodavlos and V. Schlue, On "hard stars" in general relativity, Annales Henri Poincaré, 20 (2019), 2135-2172.  doi: 10.1007/s00023-019-00793-4.

[21]

R. Geroch and L. Lindblom, Causal theories of dissipative relativistic fluids, Ann. Physics, 207 (1991), 394-416.  doi: 10.1016/0003-4916(91)90063-E.

[22]

D. Ginsberg, A priori estimates for a relativistic liquid with free surface boundary, J. Hyperbolic Differ. Equ., 16 (2019), 401-442.  doi: 10.1142/S0219891619500152.

[23]

M. Hadžić, S. Shkoller and J. Speck, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, Commun. Partial Differ. Equ., 44 (2019), 859-906.  doi: 10.1080/03605302.2019.1583250.

[24]

U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Ann. Rev. Nucl. Part. Sci., 63 (2013), 123-151. 

[25]

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

[26]

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.

[27]

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

[28]

L. Lehner, O. A. Reula and M. E. Rubio, Hyperbolic theory of relativistic conformal dissipative fluids, Phys. Rev. D, 97 (2018), 024013. doi: 10.1103/physrevd. 97.024013.

[29]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. III, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 183.

[30]

T. A. Oliynyk, Dynamical relativistic liquid bodies I: constraint propagation, preprint, arXiv: 1707.08219.

[31]

T. A. Oliynyk, A priori estimates for relativistic liquid bodies, Bull. Sci. Math., 141 (2017), 105-222.  doi: 10.1016/j.bulsci.2017.02.001.

[32]

T. A. Oliynyk, Dynamical relativistic liquid bodies, preprint, arXiv: 1907.08192.

[33] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, New York, 2013. 
[34]

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

[35]

M. Strickland, Anisotropic hydrodynamics: Motivation and methodology, Nucl. Phys. A, 926 (2014), 92-101.  doi: 10.5506/APhysPolB.45.2355.

[36]

M. Strickland, Anisotropic hydrodynamics: Three lectures, Acta Phys. Polon. B, 45 (2014), 2355-2394.  doi: 10.5506/APhysPolB.45.2355.

[37]

M. E. Taylor, Partial Differential Equations III: Nonlinear Equation, Spring, New York, 2010. doi: 10.1007/978-1-4419-7049-7.

[38] S. Weinberg, Cosmology, Oxford University Press, 2008. 
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