Ill/well-posedness of non-diffusive active scalar equations with physical applications

TL;DR

This paper investigates the conditions under which certain non-diffusive active scalar equations are well-posed or ill-posed, providing a comprehensive analysis based on the singularity order of the constitutive operator.

Contribution

It establishes new well-posedness and ill-posedness results for a broad class of non-diffusive active scalar equations depending on the operator's singularity.

Findings

Well-posedness in Gevrey spaces for $r_0\in(0,1]$

Ill-posedness for $r_0\in[1,2]$ under certain conditions

Application to specific equations like magnetogeostrophic and porous media equations

Abstract

We consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator $\mathbf{T}$ that is singular of order $r_0\in[0,2]$. For $r_0\in(0,1]$ we prove well-posedness in Gevrey spaces $G^s$ with $s\in[1,\frac{1}{r_0})$, while for $r_0\in[1,2]$ and further conditions on $\mathbf{T}$ we prove ill-posedness in $G^s$ for suitable $s$. We then apply the ill/well-posedness results to several specific non-diffusive active scalar equations including the magnetogeostrophic equation, the incompressible porous media equation and the singular incompressible porous media equation.