Non-mixing Lagrangian solutions to the multispecies Porous Media Equation

In this paper, we consider a multispecies variant of the porous media equation used in cancer modelling. Here each species represents a different cell population e.g. healthy cells versus tumor cells. The density of each population solves a continuity equation where the velocity field is given by a pressure gradient that is induced by the total cell density. The resulting model is a challenging system of coupled hyperbolic and parabolic equations, indeed, the individual species do not regularize over time and discontinuities can both form and persist. As a result, the existence of weak solutions to this model has only been achieved recently and many important questions still remain. A particularly important open question is whether it is possible for the different populations to mix together if they were separated at initial time. The main result of this paper is the construction of solutions that do not mix. To do this, we show that it is possible to construct both the forward and backward Lagrangian flows along the pressure gradient -- a result that may be of independent interest as the pressure gradient lacks sufficient regularity to apply the theory of regular Lagrangian flows. To overcome this difficulty, we exploit the structure of the porous media equation to show that the bad parts of the pressure gradient can be ignored. Once we have the flow maps, it is straightforward to show that the populations do not mix

James Floyd Ray

Capt. James Floyd Ray, August 17, 1937 - January 9, 1965 Native Sons Exhibit Pagehttps://kb.gcsu.edu/nativesons/1015/thumbnail.jp

The back-and-forth method for Wasserstein gradient flows

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced by Jacobs and L\'eger to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large-scale simulations for a large class of internal energies including singular and non-convex energies

Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoḡlu–Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations