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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Development of singularities in the relativistic Euler equations
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by Nikolaos Athanasiou, Tianrui Bayles-Rea and Shengguo Zhu PDF
Trans. Amer. Math. Soc. 376 (2023), 2325-2372 Request permission


The purpose of this paper is to study the phenomenon of singularity formation in large data problems for $C^1$ solutions to the Cauchy problem of the relativistic Euler equations. The classical theory established by [P. D. Lax [J. Math. Phys. 5 (1964), pp. 611–613] shows that, for $2\times 2$ hyperbolic systems, the break-down of $C^1$ solutions occurs in finite time if initial data contain any compression in some truly non-linear characteristic field under some additional conditions, which include genuine non-linearity and the strict positivity of the difference between two corresponding eigenvalues. These harsh structural assumptions mean that it is highly non-trivial to apply this theory to archetypal systems of conservation laws, such as the (1+1)-dimensional relativistic Euler equations. Actually, in the (1+1)-dimensional spacetime setting, if the mass-energy density $\rho$ does not vanish initially at any finite point, the essential difficulty in considering the possible break-down is coming up with a way to obtain sharp enough control on the lower bound of $\rho$. To this end, based on introducing several key artificial quantities and some elaborate analysis on the difference of the two Riemann invariants, we characterized the decay of mass-energy density lower bound in time, and ultimately made some concrete progress. On the one hand, for the $C^1$ solutions with large data and possible far field vacuum to the isentropic flow, we verified the theory obtained by P. D. Lax in 1964. On the other hand, for the $C^1$ solutions with large data and strictly positive initial mass-energy density to the non-isentropic flow, we exhibit a numerical value $N$, thought of as representing the strength of an initial compression, above which all initial data lead to a finite-time singularity formation. These singularities manifest as a blow-up in the gradient of certain Riemann invariants associated with corresponding systems.
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Additional Information
  • Nikolaos Athanasiou
  • Affiliation: American College of Thessaloniki, Thessaloniki 55535, Greece
  • MR Author ID: 1427449
  • Email:
  • Tianrui Bayles-Rea
  • Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
  • Email:
  • Shengguo Zhu
  • Affiliation: School of Mathematical Sciences, CMA-Shanghai, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China; and Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
  • ORCID: 0000-0003-3850-1727
  • Email:
  • Received by editor(s): July 22, 2021
  • Received by editor(s) in revised form: January 3, 2022
  • Published electronically: January 18, 2023
  • Additional Notes: The research was supported in part by National Key R&D Program of China (No. 2022YFA1007300), the Royal Society– Newton International Fellowships NF170015, Newton International Fellowships Alumni AL/201021 and AL/211005, and the UK Engineering and Physical Sciences Research Council Award EP/L015811/1. Most of the research work from the first author in this paper was carried out while he was based at Imperial College London. The research of the third author was also supported in part by the National Natural Science Foundation of China under Grants 12101395 and 12161141004
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 2325-2372
  • MSC (2020): Primary 83A05, 35Q31, 35Q75, 35A09, 35L67
  • DOI:
  • MathSciNet review: 4557867