On a class of forced active scalar equations with small diffusive parameters

TL;DR

This paper investigates a broad class of active scalar equations with small diffusive parameters, analyzing their well-posedness, regularity, and long-term behavior, including the effects of vanishing diffusion limits.

Contribution

It provides new mathematical insights into the behavior of active scalar equations with multiple small parameters, extending understanding of their solutions and limits.

Findings

Established well-posedness under certain conditions

Analyzed regularity properties of solutions

Explored long-time behavior and diffusion limits

Abstract

Many equations that model fluid behaviour are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form appropriate to the physics of the situation, the resulting partial differential equations often contain several small parameters. We study a general class of such PDEs called active scalar equations which in specific parameter regimes produce certain well known models for fluid motion. We address various mathematical questions relating to well-posedness, regularity and long time behaviour of the solutions to this general class including vanishing limits of several diffusive parameters.