![Illustration of causality. J-(P)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^-(P)$$\end{document} is the past light cone with vertex at P. Points inside J-(P)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^-(P)$$\end{document} can be joined to a point P in space-time by a causal past-directed curve (e.g., the red line). The value of solutions at P depends only on the data on J-(P)∩Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^-(P) \cap \Sigma $$\end{document}](https://www.researchgate.net/publication/370774840/figure/fig1/AS:11431281172570132@1688609031670/Illustration-of-causality-J-Pdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![Illustration of the space-time regions of Theorem 3. For each time t, the solution remains in equilibrium outside the ball of radius R(t)=ct+R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(t) = ct+R_0$$\end{document}. Our illustration is in 1+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+1$$\end{document} dimensions for simplicity, but the result holds in 3+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3+1$$\end{document} dimensions](https://www.researchgate.net/publication/370774840/figure/fig2/AS:11431281172579303@1688609031740/Illustration-of-the-space-time-regions-of-Theorem-3-For-each-time-t-the-solution_Q320.jpg)
![Illustration of the initial data in Theorem 3. The figure illustrates |u|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \textbf{u}\vert $$\end{document}. The velocity is zero, corresponding to equilibrium, outside the ball of radius R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}. Note that the figure shows a radially symmetric configuration for simplicity. The class of initial profiles for u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{u}$$\end{document} leading to breakdown is characterized by an integral inequality and does not require any symmetry](https://www.researchgate.net/publication/370774840/figure/fig3/AS:11431281172596026@1688609031845/Illustration-of-the-initial-data-in-Theorem-3-The-figure-illustrates_Q320.jpg)
May 2023
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70 Reads
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11 Citations
Letters in Mathematical Physics
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.
