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Annales Henri Poincaré

, Volume 20, Issue 7, pp 2173–2270 | Cite as

The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

  • Marcelo M. DisconziEmail author
  • Jared Speck
Article
  • 53 Downloads

Abstract

We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and transport-div-curl equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular, for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.

Mathematics Subject Classification

35Q75 Secondary 35L10 35Q35 35L67 

Notes

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Volume 140, 2nd edn. (Pure and Applied Mathematics). Cambridge University Press, Academic Press (2003)Google Scholar
  2. 2.
    Alinhac, S.: Blowup of small data solutions for a quasilinear wave equation in two space dimensions. Ann. Math. (2) 149(1), 97–127 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alinhac, S.: The null condition for quasilinear wave equations in two space dimensions. II. Am. J. Math. 123(6), 1071–1101 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids. EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich (2007)Google Scholar
  5. 5.
    Christodoulou, D.: The Shock Development Problem. ArXiv e-prints (May 2017). Available at arXiv:1705.00828 (2017)
  6. 6.
    Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  7. 7.
    Christodoulou, D., Lisibach, A.: Shock development in spherical symmetry. Ann. PDE 2(1), 1–246 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Christodoulou, D., Miao, S.: Compressible Flow and Euler’s Equations. Surveys of Modern Mathematics, vol. 9. International Press, Somerville (2014)Google Scholar
  9. 9.
    Coutand, D., Lindblad, H., Shkoller, S.: A priori estimates for the free boundary 3D compressible Euler equations in physical vacuum. Commun. Math. Phys. 296(2), 559–587 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum. Commun. Pure Appl. Math. 64(3), 328–366 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for the moving-boundary three dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. 206(2), 515–616 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Disconzi, M.M., Ebin, D.G.: Motion of slightly compressible fluids in a bounded domain, II. Commun. Contemp. Math. 19(4), 1650054, 57 (2017)Google Scholar
  13. 13.
    Hadžić, M., Shkoller, S., Speck, J.: A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary. Commun. Partial Differ. Equ. preprint available (November 2015). Available at arXiv:1511.07467 (2015)
  14. 14.
    Holzegel, G., Klainerman, S., Speck, J., Wong, W.W.-Y.: Small-data shock formation in solutions to 3D quasilinear wave equations: an overview. J. Hyperb. Differ. Equ. 13(01), 1–105 (2016).  https://doi.org/10.1142/S0219891616500016 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Jang, J., LeFloch, P.G., Masmoudi, N.: Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum. J. Differ. Equ. 260(6), 5481–5509 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math. 27, 377–405 (1974)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kato, T.: Linear evolution equations of “Hyperbolic” type. J. Fac. Sci. Sect. Univ. Tokyo I(17), 241–258 (1970)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kato, T.: Linear evolution equations of “Hyperbolic” type II. J. Math. Soc. Jpn. 25, 648–666 (1973)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58(3), 181–205 (1975)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Klainerman, S., Rodnianski, I.: Improved local well-posedness for quasilinear wave equations in dimension three. Duke Math. J. 117(1), 1–124 (2003)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Klainerman, S., Rodnianski, I., Szeftel, J.: The bounded \(L^{2}\) curvature conjecture. Invent. Math. 202(1), 91–216 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Luk, J., Speck, J.: The hidden null structure of the compressible Euler equations and a prelude to applications. J. Hyperb. Differ. Equ. preprint available (October 2016). Available at arXiv:1610.00743 (2016)
  23. 23.
    Luk, J., Speck, J.: Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity. Invent. Math 214(1), 1–169 (2018)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Miao, S.: On the formation of shock for quasilinear wave equations with weak intensity pulse. Ann. PDE 4(1), 140 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Miao, S., Pin, Y.: On the formation of shocks for quasilinear wave equations. Invent. Math. 207(2), 697–831 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Palais, R.S.: Seminar on the Atiyah–Singer index theorem, with contributions by M.F. Atiyah, A. Borel, E.E. Floyd, R.T. Seeley, W. Shih and R. Solovay. Annals of Mathematics Studies, No. 57. Princeton University Press, Princeton (1965)Google Scholar
  27. 27.
    Rendall, A.D.: The initial value problem for a class of general relativistic fluid bodies. J. Math. Phys. 33(3), 1047–1053 (1992)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rezzolla, L., Zanotti, O.: Relativistic Hydrodynamics. Oxford University Press, Oxford (2013)zbMATHGoogle Scholar
  29. 29.
    Sbierski, J.: On the existence of a maximal Cauchy development for the Einstein equations: a dezornification. Ann. Henri Poincaré 17(2), 301–329 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Smith, H.F., Tataru, D.: Sharp local well-posedness results for the nonlinear wave equation. Ann. Math. (2) 162(1), 291–366 (2005)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Speck, J.: Well-posedness for the Euler–Nordström system with cosmological constant. J. Hyperbolic Differ. Equ. 6(2), 313–358 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Speck, J.: Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations. Mathematical Surveys and Monographs (2016)Google Scholar
  33. 33.
    Speck, J.: A new formulation of the 3D compressible Euler equations with dynamic entropy: remarkable null structures and regularity properties. ArXiv e-prints (January 2017). Available at arXiv:1701.06626 (2017)
  34. 34.
    Speck, J.: Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation. ArXiv e-prints (September 2017). Available at arXiv:1709.04509 (2017)
  35. 35.
    Speck, J.: A Summary of Some New Results on the Formation of Shocks in the Presence of Vorticity. Nonlinear Analysis in Geometry and Applied Mathematics, pp. 133–157 (2017)Google Scholar
  36. 36.
    Speck, J.: Shock formation for 2D quasilinear wave systems featuring multiple speeds: blowup for the fastest wave, with non-trivial interactions up to the singularity. Ann. PDE 4(1), 131 (2018)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Speck, J., Holzegel, G., Luk, J., Wong, W.: Stable shock formation for nearly simple outgoing plane symmetric waves. Ann. PDE 2(2), 1–198 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Taylor, M.E.: Partial Differential Equations III: Nonlinear Equation. Springer, New York (2010)Google Scholar
  39. 39.
    Wang, Q.: A geometric approach for sharp local well-posedness of quasilinear wave equations. Ann. PDE 3(1), 12 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Weinberg, S.: Cosmology. Oxford University Press, Oxford (2008)zbMATHGoogle Scholar
  41. 41.
    Wong, W.W.-Y.: A comment on the construction of the maximal globally hyperbolic Cauchy development. J. Math. Phys. 54(11), 113511, 8 (2013)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Vanderbilt UniversityNashvilleUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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