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

Susan Friedlander; Anthony Suen; Fei Wang

paper

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

arXiv:2212.01835

Abstract

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.