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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system
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by Mohammed Lemou, Florian Méhats and Pierre Raphaël PDF
J. Amer. Math. Soc. 21 (2008), 1019-1063 Request permission

Abstract:

The three dimensional gravitational Vlasov-Poisson system $\partial _tf+v\cdot \nabla _x f-E_f\cdot \nabla _vf=0$, where $E_f(x)=\nabla _x \phi _f(x)$, $\Delta _x\phi _f=\rho _f(x)$, $\rho _f(x)=\int _{\mathbb {R}^N} f(x,v)dv$, describes the mechanical state of a stellar system subject to its own gravity. Smooth initial data yield unique global in time solutions from a celebrated result by Pfaffelmoser. There exists a hierarchy of physical models which aim at taking into account further relativistic effects. The simplest one is the three dimensional relativistic gravitational Vlasov-Poisson system $\partial _tf+\frac {v}{\sqrt {1+|v|^2}}\cdot \nabla _x f-E_f\cdot \nabla _vf=0$ which we study here. A striking feature as observed by Glassey and Schaeffer is that this system now admits finite blow up solutions. Nevertheless, the existence argument is purely obstructive and provides no insight into the description of the singularity formation. We exhibit in this paper a family of finite time blow up self-similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations. Our analysis applies to the four dimensional gravitational Vlasov-Poisson system as well.
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Additional Information
  • Mohammed Lemou
  • Affiliation: CNRS and Université Paul Sabatier, MIP, 118, Route de Narbonne, 31062 Toulouse, France
  • MR Author ID: 355223
  • Florian Méhats
  • Affiliation: IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
  • MR Author ID: 601414
  • Pierre Raphaël
  • Affiliation: CNRS and Université Paris-Sud, Orsay, France
  • Received by editor(s): November 8, 2006
  • Published electronically: November 29, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 21 (2008), 1019-1063
  • MSC (2000): Primary 82C70, 35Q55, 35Q75, 85A05, 74H35
  • DOI: https://doi.org/10.1090/S0894-0347-07-00579-6
  • MathSciNet review: 2425179