Lagrangian solutions to the Porous Media Equation and Reaction Diffusion Systems

In this paper, we construct global-in-time forward and backward Lagrangian flow maps along the pressure gradient generated by weak solutions of the Porous Media Equation. The main difficulty is that when the initial data has compact support, 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 the flow maps. To overcome this difficulty, we develop a new argument that combines Aronson-B\'enilan type estimates with the quantitative Lagrangian flow theory of Crippa and De Lellis to show that certain doubly logarithmic quantities measuring the stability of flow maps do not blow up fast enough to prevent compactness. Our arguments are sufficiently flexible to handle the Hele-Shaw limit and a multispecies generalization of the Porous Media Equation where the equation is replaced by a coupled hyperbolic-parabolic system of reaction diffusion equations. As one application of our flow maps, we are able to construct solutions where different species cannot mix together if they were separated at initial time.