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

2022/12/04 by Friedlander, Susan, Suen, Anthony, Wang, Fei
#35Q35 #76D03 #76W05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/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 r₀∈[0,2]. For r₀∈(0,1] we prove well-posedness in Gevrey spaces G^s with s∈[1,(1)/(r₀)), 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.

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