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Breakdown of smooth solutions to the Müller–Israel–Stewart equations of relativistic viscous fluids

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Abstract

We consider equations of Müller–Israel–Stewart type describing a relativistic viscous fluid with bulk viscosity in four-dimensional Minkowski space. We show that there exists a class of smooth initial data that are localized perturbations of constant states for which the corresponding unique solutions to the Cauchy problem break down in finite time. Specifically, we prove that in finite time such solutions develop a singularity or become unphysical in a sense that we make precise. We also show that in general Riemann invariants do not exist in 1+1 dimensions for physically relevant equations of state and viscosity coefficients. Finally, we present a more general version of a result by Y. Guo and A.S. Tahvildar-Zadeh: We prove large-data singularity formation results for perfect fluids under very general assumptions on the equation of state, allowing any value for the fluid sound speed strictly less than the speed of light.

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Acknowledgements

The authors would like to thank Jorge Noronha for useful discussions on a preliminary version of this manuscript. MMD gratefully acknowledges support from NSF grant # 2107701, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, from a Discovery grant administered by Vanderbilt University, and from a Deans’ Faculty Fellowship. VH’s work on this project was funded (full or in-part) by the University of Texas at San Antonio, Office of the Vice President for Research, Economic Development, and Knowledge Enterprise. VH gratefully acknowledges partial support by NSF grants DMS-1614797 and DMS-1810687.

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  1. Vu Hoang and Maria Radosz have contributed equally to this work.

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  1. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center Ln, Nashville, TN, 37212, USA

    Marcelo M. Disconzi

  2. Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USA

    Vu Hoang & Maria Radosz

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Disconzi, M.M., Hoang, V. & Radosz, M. Breakdown of smooth solutions to the Müller–Israel–Stewart equations of relativistic viscous fluids. Lett Math Phys 113, 55 (2023). https://doi.org/10.1007/s11005-023-01677-9

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