On the regularity of the solution map of the incompressible porous media equation

In this paper, we consider the incompressible porous media equation on the Sobolev space Hs(R2),s>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s({\mathbb {R}}^2),\;s > 2$$\end{document}. We provide a Lagrangian formulation of this equation on the Sobolev-type diffeomorphism group Ds(R2),s>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}^s({\mathbb {R}}^2),\;s > 2$$\end{document}. It turns out that this Lagrangian formulation generates an analytic dynamics. This analyticity in the Lagrangian picture will immediately lead to the result that the particle trajectories of the incompressible porous media flow are analytic curves in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^2$$\end{document}. In the Eulerian picture, the situation is drastically different. We prove that for T>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T > 0$$\end{document}, the time T solution map of the incompressible porous media equation ρ0↦ρ(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0 \mapsto \rho (T)$$\end{document}, mapping the initial value of the solution to its time T value, is nowhere locally uniformly continuous and hence nowhere locally Lipschitz.