Remarks on Lifespan and Continuation Criteria of Two Dimensional Incompressible Fluid Models

TL;DR

This paper investigates the lifespan and continuation conditions of 2D incompressible fluid models, introducing a novel energy-vorticity approach to establish long-time existence results and a new BKM-type criterion.

Contribution

It presents a new energy-vorticity formulation and bootstrap method to prove long-time existence and a conditional criterion for 2D fluid models near Euler conditions.

Findings

Established several long-time existence results for 2D fluid models.

Derived a new conditional BKM-type criterion for inhomogeneous Euler equations.

Developed a novel energy-vorticity formulation combining linear transport and bootstrap techniques.

Abstract

In this paper, we study the lifespan and continuation criteria of several two-dimensional incompressible fluid models. Motivated by a novel energy-vorticity formulation, combining linear transport estimate and a bootstrap argument, we are able to show several longtime existence results of two dimensional fluid models in the close-to-Euler regime. A byproduct of our approach is a new conditional BKM type result for the inhomogeneous incompressible Euler equation.