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

We consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator 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 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.