Non-radial implosion for compressible Euler and Navier-Stokes in $\mathbb{T}^3$ and $\mathbb{R}^3$

TL;DR

This paper constructs smooth, non-radial solutions to the compressible Euler and Navier-Stokes equations that develop finite-time singularities, extending previous work to more general initial data in both periodic and non-radial settings.

Contribution

It introduces a flexible method to construct non-radial solutions with finite-time singularities for compressible fluid equations, broadening the scope beyond radial symmetry.

Findings

Existence of smooth solutions that blow up in finite time

Applicable to both periodic and non-radial initial data

Extends previous radial symmetry results

Abstract

In this paper we construct smooth, non-radial solutions of the compressible Euler and Navier-Stokes equation that develop an imploding finite time singularity. Our construction is motivated by the works [Merle, Rapha\"{e}l, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022], [Buckmaster, Cao-Labora, and G\'{o}mez-Serrano, arXiv:2208.09445, 2022], but is flexible enough to handle both periodic and non-radial initial data.