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In this paper, we construct global-in-time forward and backward Lagrangianflow maps along the pressure gradient generated by weak solutions of the PorousMedia Equation. The main difficulty is that when the initial data has compactsupport, it is well-known that the pressure gradient is not a BV function.Thus, the theory of regular Lagrangian flows cannot be applied to construct theflow maps. To overcome this difficulty, we develop a new argument that combinesAronson-B\'enilan type estimates with the quantitative Lagrangian flow theoryof Crippa and De Lellis to show that certain doubly logarithmic quantitiesmeasuring the stability of flow maps do not blow up fast enough to preventcompactness. Our arguments are sufficiently flexible to handle the Hele-Shawlimit and a multispecies generalization of the Porous Media Equation where theequation is replaced by a coupled hyperbolic-parabolic system of reactiondiffusion equations. As one application of our flow maps, we are able toconstruct solutions where different species cannot mix together if they wereseparated at initial time.

Lagrangian solutions to the Porous Media Equation and Reaction Diffusion Systems