Blow up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations

TL;DR

This paper proves that smooth solutions to the compressible Navier-Stokes equations in three or more dimensions inevitably develop singularities in finite time, regardless of initial data support.

Contribution

It establishes finite-time blow-up of smooth solutions for the compressible Navier-Stokes equations without requiring compact initial support.

Findings

Smooth solutions lose regularity in finite time in 3D or higher.

Blow-up occurs even with non-compact initial data.

Results hold under conserved mass, energy, and finite momentum of inertia.

Abstract

We prove that the smooth solutions to the Cauchy problem for the Navier-Stokes equations with conserved mass, total energy and finite momentum of inertia loses the initial smoothness within a finite time in the case of space of dimension 3 or greater even if the initial data are not compactly supported.