September  2021, 20(9): 2885-2914. doi: 10.3934/cpaa.2021068

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

1. 

Universidade Federal do Rio Grande do Norte, Campus Universitário Lagoa Nova, Natal, RN, 59078-970, Brazil

2. 

Vanderbilt University, 1326 Stevenson Center Ln, Nashville, TN, 37240, USA

3. 

Baylor University, Sid Richardson Building, 1410 S. 4th Street, Waco, TX 76706, USA

* Corresponding author

Received  August 2020 Revised  March 2021 Published  September 2021 Early access  June 2021

Fund Project: 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

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.

Citation: Fabio S. Bemfica, Marcelo M. Disconzi, P. Jameson Graber. Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2885-2914. doi: 10.3934/cpaa.2021068
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show all references

References:
[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.

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[4]

B. Abbott et al., GW170817: Measurements of neutron star radii and equation of state, Phys. Rev. Lett., 121 (2018), 161101.

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L. Adamczyk et al., Global $\Lambda$ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature, 548 (2017), 62-65. 

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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.

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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.

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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.

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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.

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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.

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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.

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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]

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