On the behaviors of solution near possible blow-up time in the incompressible Euler and related equations

TL;DR

This paper investigates the behavior of solutions near potential blow-up times in incompressible Euler, Navier-Stokes, and related equations, deriving new blow-up criteria and necessary conditions for singularity formation.

Contribution

It introduces new blow-up criteria and necessary conditions for solutions to blow up in finite time for several fundamental fluid dynamics equations.

Findings

Derived possible blow-up behaviors of scalar quantities.

Established new blow-up criteria for Euler and related equations.

Identified necessary conditions for finite-time blow-up.

Abstract

We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier-Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of the scaling invariant norms, we derive the possible blow-up behaviors of the above quantities, from which we obtain new type of blow-up criteria and some necessary conditions for the blow-up.