Remarks on the asymptotically discretely self-similar solutions of the Navier-Stokes and the Euler equations

TL;DR

This paper investigates the possibility of self-similar blow-up solutions in 3D Navier-Stokes and Euler equations, proving non-existence under certain regularity and integrability conditions for these solutions.

Contribution

It generalizes previous notions of self-similarity to locally asymptotically discretely self-similar solutions and establishes their non-existence under specific conditions.

Findings

No locally asymptotically discretely self-similar blow-up for Navier-Stokes with periodic profiles.

Excludes asymptotically discretely self-similar blow-up for Euler under integrability conditions.

Provides conditions under which self-similar blow-up solutions cannot occur.

Abstract

We study scenarios of self-similar type blow-up for the incompressible Navier-Stokes and the Euler equations. The previous notions of the discretely (backward) self-similar solution and the asymptotically self-similar solution are generalized to the locally asymptotically discretely self-similar solution. We prove that there exists no such locally asymptotically discretely self-similar blow-up for the 3D Navier-Stokes equations if the blow-up profile is a time periodic function belonging to $C^1(\Bbb R ; L^3(\Bbb R^3)\cap C^2 (\Bbb R^3))$. For the 3D Euler equations we show that the scenario of asymptotically discretely self-similar blow-up is excluded if the blow-up profile satisfies suitable integrability conditions.