Backward self-similar solutions for compressible Navier-Stokes equations

TL;DR

This paper investigates the existence of backward self-similar blow-up solutions for the compressible Navier-Stokes equations with radial symmetry, concluding they cannot exist under certain conditions and conjecturing their non-existence generally.

Contribution

The paper proves non-existence results for backward self-similar solutions under smallness and finite energy conditions, and supports the conjecture through numerical simulations.

Findings

Backward self-similar solutions do not exist under smallness conditions.

Finite energy solutions of this type do not exist.

Numerical simulations support the conjecture of non-existence.

Abstract

This article is devoted to backward self-similar blow up solutions of the compressible Navier-Stokes equations with radial symmetry. We show that such solutions cannot exist if they either satisfy an appropriate smallness condition, or have finite energy. Furthermore, numerical simulations lead us to the conjecture that such solutions do not exist.