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Fractional oscillon equations; solvability and connection with classical oscillon equations
Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics
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 |
In this manuscript, we study the theory of conformal relativistic viscous hydrodynamics introduced in [
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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