On the implosion of a three dimensional compressible fluid

2019/12/23 by Frank Merle, Pierre Raphaël, Merle, Frank +5 · 2 citations
Earth and Planetary Sciences · Engineering · Mathematics · #35Q35 #Analysis of PDEs (math.AP) #Aquatic and Environmental Studies #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1912.11009

openalex publication_date 2019/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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. Two essential steps of the analysis are the existence of \mathcal C^∞ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow which are addressed in the companion papers \citeMRRSprofile, MRRSdefoc. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).

Cited by

• Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation
• Magnetic Stabilization of Compressible Flows: Global Existence in 3D Inviscid Non-Isentropic MHD Equations

Related

• Smooth self-similar imploding profiles to 3D compressible Euler
• Non-radial implosion for compressible Euler and Navier-Stokes in \mathbbT³ and ℝ³
• On smooth self similar solutions to the compressible Euler equations
• Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases
• New Self-similar Euler Flows: gradient catastrophe without shock formation