Abstract
We study the singularity formation of smooth solutions of the relativistic Euler equations in (3 + 1)-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial ``generalized'' momentum is sufficiently large.
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Communicated by G.W. Gibbons
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Pan, R., Smoller, J. Blowup of Smooth Solutions for Relativistic Euler Equations. Commun. Math. Phys. 262, 729–755 (2006). https://doi.org/10.1007/s00220-005-1464-9
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DOI: https://doi.org/10.1007/s00220-005-1464-9