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

Susan Friedlander; Anthony Suen; Fei Wang

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arXiv · 2212.01835

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

Susan Friedlander·Anthony Suen·Fei Wang

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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.

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Susan Friedlander, Anthony Suen, Fei Wang. 2022-12-04. Ill/well-posedness of non-diffusive active scalar equations with physical applications. https://arxiv.org/abs/2212.01835

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