On the implosion of a compressible fluid II:Singularity formation

Frank Merle; Pierre Raphaël; Igor Rodnianski; Jeremie Szeftel

doi:10.4007/ANNALS.2022.196.2.4

On the implosion of a compressible fluid II:Singularity formation

Frank Merle, Pierre Raphaël, Igor Rodnianski, Jeremie Szeftel

Mathematics

Princeton University

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is the existence of (Formula Presented) smooth selfsimilar solutions to the compressible Euler equations for quantized values of the speed constructed in our companion paper (part I).

Original language	American English
Pages (from-to)	779-889
Number of pages	111
Journal	Annals of Mathematics
Volume	196
Issue number	2
DOIs	https://doi.org/10.4007/ANNALS.2022.196.2.4
State	Published - Sep 2022

ASJC Scopus subject areas

Mathematics (miscellaneous)

Keywords

Compressible fluids
Singularity formation

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10.4007/ANNALS.2022.196.2.4

Cite this

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abstract = "In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is the existence of (Formula Presented) smooth selfsimilar solutions to the compressible Euler equations for quantized values of the speed constructed in our companion paper (part I).",

keywords = "Compressible fluids, Singularity formation",

author = "Frank Merle and Pierre Rapha{\"e}l and Igor Rodnianski and Jeremie Szeftel",

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AU - Merle, Frank

AU - Raphaël, Pierre

AU - Rodnianski, Igor

AU - Szeftel, Jeremie

PY - 2022/9

Y1 - 2022/9

N2 - In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is the existence of (Formula Presented) smooth selfsimilar solutions to the compressible Euler equations for quantized values of the speed constructed in our companion paper (part I).

AB - In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is the existence of (Formula Presented) smooth selfsimilar solutions to the compressible Euler equations for quantized values of the speed constructed in our companion paper (part I).

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KW - Singularity formation

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On the implosion of a compressible fluid II:Singularity formation

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Keywords

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