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Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics

  • * Corresponding author

    * Corresponding author 
The second author is partially supported by a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, by NSF grant DMS-1812826, and by a Discovery grant administered by Vanderbilt University, and from a Dean's Faculty Fellowship. The third author is partially supported by NSF grant DMS-1905449
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  • We study the theory of relativistic viscous hydrodynamics introduced in [14, 58], which provided a causal and stable first-order theory of relativistic fluids with viscosity in the case of barotropic fluids. The local well-posedness of its equations of motion has been previously established in Gevrey spaces. Here, we improve this result by proving local well-posedness in Sobolev spaces.

    Mathematics Subject Classification: Primary: 35Q75; Secondary: 35Q35.


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  • [1] B. Abbott et al., Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, Astrophys. J. Lett., 848 (2017), L13.
    [2] B. Abbott et al., GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett., 119 (2017), 161101.
    [3] B. Abbott et al., Multi-messenger Observations of a Binary Neutron Star Merger, Astrophys. J. Lett., 848 (2017), L12.
    [4] B. Abbott et al., GW170817: Measurements of neutron star radii and equation of state, Phys. Rev. Lett., 121 (2018), 161101.
    [5] L. Adamczyk et al., Global $\Lambda$ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature, 548 (2017), 62-65. 
    [6] 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.
    [7] K. Allen, A. Collins, J. Maderer, S. Perus and E. Velasco, LIGO and Virgo make first detection of gravitational waves produced by colliding neutron stars, LIGO-Virgo Press Release, http://www.ligo.org/detections/GW170817/press-release/pr-english.pdf.
    [8] M. AlqahtaniM. Nopoush and M. Strickland, Relativistic anisotropic hydrodynamics, Prog. Part. Nucl. Phys., 101 (2018), 204-248. 
    [9] A. M. AnileRelativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics (Cambridge Monographs on Mathematical Physics), Cambridge University Press, 1 edition, 1990. 
    [10] 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.
    [11] F. S. Bemfica, M. M. Disconzi, V. Hoang, J. Noronha and M. Radosz, Nonlinear Constraints on Relativistic Fluids Far From Equilibrium, arXiv: 2005.11632.
    [12] 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, 26. doi: 10.1103/physrevd. 98.104064.
    [13] 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.
    [14] F. S. Bemfica, M. M. Disconzi and J. Noronha, Nonlinear causality of general first-order relativistic viscous hydrodynamics, Phys. Rev. D, 100 (2019), 104020, 13. doi: 10.1103/physrevd. 100.104020.
    [15] F. S. Bemfica, M. M. Disconzi, C. Rodriguez and Y. Shao, Local well-posedness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics, arXiv: 1911.02504.
    [16] S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP, 02 (2008), 045.
    [17] C. H. ChanM. Czubak and M. M. Disconzi, The formulation of the Navier-Stokes equations on Riemannian manifolds, J. Geom. Phys., 121 (2017), 335-346.  doi: 10.1016/j.geomphys.2017.07.015.
    [18] E. A. Chodos, Looking back: The top ten physics newsmakers of the decade, APS News, 19, https://www.aps.org/publications/apsnews/201002/newsmakers.cfm.
    [19] Y. Choquet-Bruhat, Diagonalisation des systèmes quasi-linéaires et hyperbolicité non stricte, J. Math. Pures Appl., 45 (1966), 371-386. 
    [20] Y. Choquet-BruhatGeneral Relativity and the Einstein Equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2009. 
    [21] D. Christodoulou, The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/031.
    [22] D. Christodoulou, The Shock Development Problem, European Mathematical Society (EMS), Zürich, 2019. doi: 10.4171/192.
    [23] M. Czubak and M. M. Disconzi, On the well-posedness of relativistic viscous fluids with non-zero vorticity, J. Math. Phys., 57 (2016), 042501, 21. doi: 10.1063/1.4944910.
    [24] S. De Groot, Relativistic Kinetic Theory, Principles and Applications, 1980.,
    [25] G. S. Denicol and J. Noronha, Divergence of the Chapman-Enskog expansion in relativistic kinetic theory, arXiv: 1608.07869.
    [26] G. Denicol, T. Kodama, T. Koide and P. Mota, Stability and Causality in relativistic dissipative hydrodynamics, J. Phys. G, 35 (2008), 115102.
    [27] G. Denicol, H. Niemi, E. Molnar and D. Rischke, Derivation of transient relativistic fluid dynamics from the Boltzmann equation, Phys. Rev. D, 85 (2012), 114047 doi: 10.1103/PhysRevD. 93.114025.
    [28] M. M. Disconzi, On the well-posedness of relativistic viscous fluids, Nonlinearity, 27 (2014), 1915-1935.  doi: 10.1088/0951-7715/27/8/1915.
    [29] 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.
    [30] M. M. Disconzi, V. Hoang and M. Radosz, Breakdown of smooth solutions to the Müller-Israel-Stewart equations of relativistic viscous fluids, arXiv: 2008.03841.
    [31] M. M. Disconzi, M. Ifrim and D. Tataru, The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion, arXiv: 2007.05787.
    [32] M. M. Disconzi, T. W. Kephart and R. J. Scherrer, New approach to cosmological bulk viscosity, Phys. Rev. D, 91 (2015), 043532, 6. doi: 10.1103/PhysRevD. 91.043532.
    [33] 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, Internat. J. Modern Phys. D, 26 (2017), 1750146, 52. doi: 10.1142/S0218271817501462.
    [34] M. M. Disconzi and J. Speck, The relativistic Euler equations: remarkable null structures and regularity properties, Ann. Henri Poincaré, 20 (2019), 2173-2270.  doi: 10.1007/s00023-019-00801-7.
    [35] C. Eckart, The Thermodynamics of irreversible processes. 3. Relativistic theory of the simple fluid, Phys. Rev., 58 (1940), 919-924. 
    [36] A. Einstein, The formal foundation of the general theory of relativity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1914 (1914), 1030-1085. 
    [37] Y. Elskens and M. K.-H. Kiessling, Microscopic foundations of kinetic plasma theory: The relativistic vlasov-maxwell equations and their radiation-reaction-corrected generalization, J. Stat. Phys., 180 (2020), 749-772.  doi: 10.1007/s10955-020-02519-x.
    [38] S. Floerchinger and E. Grossi, Causality of fluid dynamics for high-energy nuclear collisions, JHEP, 08 (2018), 186. doi: 10.1007/jhep08(2018)186.
    [39] Y. Fourès-Bruhat, Théorèmes d'existence en mécanique des fluides relativistes, Bull. Soc. Math. France, 86 (1958), 155-175. 
    [40] H. Freistühler, A class of Hadamard well-posed five-field theories of dissipative relativistic fluid dynamics, J. Math. Phys., 61 (2020), 033101, 17pp. doi: 10.1063/1.5135704.
    [41] H. Freistühler and B. Temple, Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation, Proc. R. Soc. Lond. Ser. A, 470 (2014), 20140055, 17. doi: 10.1098/rspa. 2014.0055.
    [42] H. Freistühler and B. Temple, Causal dissipation for the relativistic dynamics of ideal gases, Proc. A., 473 (2017), 20160729, 20. doi: 10.1098/rspa. 2016.0729.
    [43] H. Freistühler and B. Temple, Causal dissipation in the relativistic dynamics of barotropic fluids, J. Math. Phys., 59 (2018), 063101, 17. doi: 10.1063/1.5007831.
    [44] R. Geroch and L. Lindblom, Causal theories of dissipative relativistic fluids, Ann. Phys., 207 (1991), 394-416.  doi: 10.1016/0003-4916(91)90063-E.
    [45] R. Geroch and L. Lindblom, Dissipative relativistic fluid theories of divergence type, Phys. Rev. D (3), 41 (1990), 1855-1861.  doi: 10.1103/PhysRevD.41.1855.
    [46] 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.
    [47] J. L. GuermondF. Marpeau and B. Popov, A fast algorithm for solving first-order PDEs by $L^1$-minimization, Commun. Math. Sci., 6 (2008), 199-216. 
    [48] M. HadžićS. 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.
    [49] U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Ann. Rev. Nucl. Part. Sci., 63 (2013), 123-151. 
    [50] M. P. Heller, A. Kurkela, M. Spaliński and V. Svensson, Hydrodynamization in kinetic theory: Transient modes and the gradient expansion, Phys. Rev. D, 97 (2018), 091503.
    [51] W. Hiscock and L. Lindblom, Stability and causality in dissipative relativistic fluids, Annals Phys., 151 (1983), 466-496.  doi: 10.1016/0003-4916(83)90288-9.
    [52] W. A. Hiscock and L. Lindblom, Generic instabilities in first-order dissipative relativistic fluid theories, Phys. Rev. D, 31 (1985), 725-733.  doi: 10.1103/PhysRevD.31.725.
    [53] R. E. Hoult and P. Kovtun, Stable and causal relativistic Navier-Stokes equations, JHEP, 06 (2020), 067. doi: 10.1007/jhep06(2020)067.
    [54] W. Israel, Nonstationary irreversible thermodynamics: A Causal relativistic theory, Annals Phys., 100 (1976), 310-331.  doi: 10.1016/0003-4916(76)90064-6.
    [55] W. Israel and J. Stewart, Transient relativistic thermodynamics and kinetic theory, Annals Phys., 118 (1979), 341-372.  doi: 10.1016/0003-4916(79)90130-1.
    [56] J. JangP. 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.
    [57] D. Jou, J. Casas-Vázquez and G. Lebon, Extended irreversible thermodynamics, 4th edition, Springer, New York, 2010. doi: 10.1007/978-90-481-3074-0.
    [58] P. Kovtun, First-order relativistic hydrodynamics is stable, JHEP, 10 (2019), 034. doi: 10.1007/jhep10(2019)034.
    [59] H.-O. KreissG. B. NagyO. E. Ortiz and O. A. Reula, Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories, J. Math. Phys., 38 (1997), 5272-5279.  doi: 10.1063/1.531940.
    [60] T. B. N. Laboratory, 'Perfect liquid' quark-gluon plasma is the most vortical fluid, Phys. org, https://phys.org/news/2017-08-liquid-quark-gluon-plasma-vortical-fluid.html.
    [61] L. D. Landau and E. Lifshitz, Fluid Mechanics - Volume 6 (Corse of Theoretical Physics), 2nd edition, Butterworth-Heinemann, 1987.
    [62] 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.
    [63] J. Leray and Y. Ohya, Équations et systèmes non-linéaires, hyperboliques nonstricts, Math. Ann., 170 (1967), 167-205.  doi: 10.1007/BF01350150.
    [64] A. Lichnerowicz, Théories Relativistes de la Gravitation et de l'Électromagnétism, Masson et Cie, Paris, 1955.
    [65] A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics: Lectures on the Existence of Solutions, W. A. Benjamin, New York, 1967. doi: 10.1007/978-94-017-2126-4.
    [66] I. S. LiuI. Müller and T. Ruggeri, Relativistic thermodynamics of gases, Ann. Phys., 169 (1986), 191-219.  doi: 10.1016/0003-4916(86)90164-8.
    [67] J. Marschall, Pseudodifferential operators with nonregular symbols of the class $S^m_{\rho\delta}$, Commun. Partial Differ. Equ., 12 (1987), 921-965.  doi: 10.1080/03605308708820514.
    [68] J. Marschall, Correction to: "Pseudodifferential operators with nonregular symbols of the class $S^m_{\rho, \delta}$", Commun. Partial Differ. Equ., 13 (1988), 129-130. 
    [69] J. Marschall, Pseudodifferential operators with coefficients in Sobolev spaces, Trans. Amer. Math. Soc., 307 (1988), 335-361.  doi: 10.2307/2000766.
    [70] S. Miao, S. Shahshahani and S. Wu, Well-posedness of the free boundary hard phase fluids in minkowski background and its newtonian limit, arXiv: 2003.02987.
    [71] E. R. Most, L. J. Papenfort, V. Dexheimer, M. Hanauske, S. Schramm, H. St'ócker and L. Rezzolla, Signatures of quark-hadron phase transitions in general-relativistic neutron-star mergers, Phys. Rev. Lett., 122 (2019), 061101.
    [72] I. Mueller, Zum Paradox der Wärmeleitungstheorie, Zeit. fur Phys, 198 (1967), 329-344. 
    [73] I. Müller and T. Ruggeri, Rational extended thermodynamics, in Springer Tracts in Natural Philosophy, 2nd edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-2210-1.
    [74] G. B. NagyO. E. Ortiz and O. A. Reula, The behavior of hyperbolic heat equations' solutions near their parabolic limits, J. Math. Phys., 35 (1994), 4334-4356.  doi: 10.1063/1.530856.
    [75] T. A. Oliynyk, On the existence of solutions to the relativistic Euler equations in two spacetime dimensions with a vacuum boundary, Classical Quantum Gravity, 29 (2012), 155013, 28. doi: 10.1088/0264-9381/29/15/155013.
    [76] T. A. Oliynyk, Dynamical relativistic liquid bodies, arXiv: 1907.08192.
    [77] T. S. Olson, Stability and causality in the Israel-Stewart energy grame theory, Ann. Phys., 199 (1990), 18. doi: 10.1016/0003-4916(90)90366-V.
    [78] T. S. Olson and W. A. Hiscock, Plane steady shock waves in Israel-Stewart fluids, Ann. Phys., 204 (1990), 331-350.  doi: 10.1016/0003-4916(90)90393-3.
    [79] J. Peralta-Ramos and E. Calzetta, Divergence-type nonlinear conformal hydrodynamics, Phys. Rev. D, 80 (2009), 126002.
    [80] J. Peralta-Ramos and E. Calzetta, Divergence-type 2+1 dissipative hydrodynamics applied to heavy-ion collisions, Phys. Rev. C, 82 (2010), 054905.
    [81] G. Pichon, Étude relativiste de fluides visqueux et chargés, Ann. Inst. H. Poincaré Sect. A (N.S.), 2 (1965), 21-85. 
    [82] S. Pu, T. Koide and D. H. Rischke, Does stability of relativistic dissipative fluid dynamics imply causality?, Phys. Rev. D, 81 (2010), 114039.
    [83] O. A. Reula and G. B. Nagy, A causal statistical family of dissipative divergence-type fluids, J. Phys. A, 30 (1997), 1695-1709.  doi: 10.1088/0305-4470/30/5/030.
    [84] L. Rezzolla and  O. ZanottiRelativistic Hydrodynamics, Oxford University Press, New York, 2013. 
    [85] P. Romatschke and  U. RomatschkeRelativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2019.  doi: 10.1017/9781108651998.
    [86] S. Ryu, J. F. Paquet, C. Shen, G. Denicol, B. Schenke, S. Jeon and C. Gale, Effects of bulk viscosity and hadronic rescattering in heavy ion collisions at energies available at the BNL Relativistic Heavy Ion Collider and at the CERN Large Hadron Collider, Phys. Rev. C, 97 (2018), 034910.
    [87] K. Schwarzschild, On the gravitational field of a sphere of incompressible fluid according to Einstein's theory, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1916 (1916), 424-434. 
    [88] M. Shibata and K. Kiuchi, Gravitational waves from remnant massive neutron stars of binary neutron star merger: Viscous hydrodynamics effects, Phys. Rev. D, 95 (2017), 123003.
    [89] M. Shibata, K. Kiuchi and Y. i. Sekiguchi, General relativistic viscous hydrodynamics of differentially rotating neutron stars, Phys. Rev. D, 95 (2017), 083005.
    [90] J. Speck and R. M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Commun. Math. Phys., 304 (2011), 229-280.  doi: 10.1007/s00220-011-1207-z.
    [91] Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition, Commun. Pure Appl. Math., 62 (2009), 1551-1594.  doi: 10.1002/cpa.20282.
    [92] K. TsumuraT. Kunihiro and K. Ohnishi, Derivation of covariant dissipative fluid dynamics in the renormalization-group method, Phys. Lett. B, 646 (2007), 134-140.  doi: 10.1016/j.physletb.2006.12.074.
    [93] K. Tsumura and T. Kunihiro, First-Principle Derivation of Stable First-Order Generic-Frame Relativistic Dissipative Hydrodynamic Equations from Kinetic Theory by Renormalization-Group Method, Prog. Theor. Phys., 126 (2011), 761-809. 
    [94] K. Tsumura and T. Kunihiro, Uniqueness of Landau-Lifshitz Energy Frame in Relativistic Dissipative Hydrodynamics, Phys. Rev. E, 87 (2013), 053008.
    [95] P. Van and T. Biro, Relativistic hydrodynamics - causality and stability, Eur. Phys. J. ST, 155 (2008), 201-212. 
    [96] P. Van and T. Biro, First order and stable relativistic dissipative hydrodynamics, Phys. Lett. B, 709 (2012), 106-110.  doi: 10.1016/j.physletb.2012.02.006.
    [97] S. WeinbergCosmology, Oxford University Press, 2008. 
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