Smooth imploding solutions for 3D compressible fluids

Building upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [67, 68, 69], we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $\gamma>1$ . For the particular case $\gamma =\frac 7...

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Bibliographic Details
Published inForum of mathematics. Pi Vol. 13
Main Authors Buckmaster, Tristan, Cao-Labora, Gonzalo, Gómez-Serrano, Javier
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 12.02.2025
Subjects
35Q30
35Q35
65G30
76N10
Differential Equations
35Q30
35Q35
76N10
65G30
Online AccessGet full text
ISSN2050-5086
DOI10.1017/fmp.2024.12

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Summary:Building upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [67, 68, 69], we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $\gamma>1$ . For the particular case $\gamma =\frac 75$ (corresponding to a diatomic gas – for example, oxygen, hydrogen, nitrogen), akin to the result [68], we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability [67] and nonlinear stability [69], which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $\gamma =\frac 75$ . Moreover, unlike [69], the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting.
ISSN:2050-5086
DOI:10.1017/fmp.2024.12