Smooth imploding solutions for 3D compressible fluids

TL;DR

This paper constructs smooth, self-similar imploding solutions for 3D compressible Euler and Navier-Stokes equations, demonstrating singularity formation with stable solutions for specific gases and providing simplified stability proofs.

Contribution

It introduces new smooth imploding solutions for 3D compressible fluids, including stability analysis and solutions for specific adiabatic exponents, advancing understanding of singularity formation.

Findings

Existence of smooth, self-similar imploding solutions for all b3>1

Construction of stable solutions for b3=7/5 in Euler and Navier-Stokes equations

Solutions have density bounded away from zero and tend to a constant at infinity

Abstract

Building upon the pioneering work [Merle, Rapha\"el, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] 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 $\gamma=\frac75$ (corresponding to a diatomic gas, e.g. oxygen, hydrogen, nitrogen), akin to the previous result, we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability and non-linear stability, which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $\gamma=\frac75$. Moreover, 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.