## Abstract

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 language | English (US) |
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Pages (from-to) | 567-778 |

Number of pages | 212 |

Journal | Annals of Mathematics |

Volume | 196 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2022 |

## All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)

## Keywords

- Euler equations
- Self-similar profile