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 r₀∈[0,2]. For r₀∈(0,1] we prove well-posedness in Gevrey spaces G^s with s∈[1, [Formula Presented]), while for r₀∈[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. Copyright © 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.