On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles

Frank Merle, Pierre Raphaël, Igor Rodnianski, Jeremie Szeftel

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21 Scopus citations


In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible threedimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension d ≥ 2 with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated acoustic cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic C∞ self-similar solutions with suitable decay at infinity.

Original languageEnglish (US)
Pages (from-to)567-778
Number of pages212
JournalAnnals of Mathematics
Issue number2
StatePublished - Sep 2022

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


  • Euler equations
  • Self-similar profile


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