JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: $L^p$ and Sobolev bounds

Abstract

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the L p and L $\infty$ norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher order results, and in particulat to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called five gradients inequality allows to recover some inequalities that can be deduced from the Bakry-Emery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller-Segel chemiotaxis model. 1 Short introduction The goal of this paper is to present some estimates on evolution PDEs in the space of probability densities which share two important features: they include a linear diffusion term, and they are gradient flows in the Wasserstein space W2. These PDEs will be of the form $\partial$t$\rho$ -- $\Delta$$\rho$ -- $\nabla$ $\times$ ($\rho$$\nabla$u[$\rho$]) = 0, complemented with no-flux boundary conditions and an intial condition on $\rho$0. We will in particular concentrate on the Fokker-Plack case, where u[$\rho$] = V and V is a fixed function (with possible regularity assumptions) independent of $\rho$, on the case where u[$\rho$] = W * $\rho$ is obtained by convolution and models interaction between particles, and on the parabolic-elliptic Keller-Segel case where u[$\rho$] is related to $\rho$ via an elliptic equation. This last case models the evolution of a biological population $\rho$ subject to diffusion but attracted by the concentration of a chemo-attractant, a nutrient which is produced by the population itself, so that its distribution is ruled by a PDE where the density $\rho$ appears as a source term. Under the assumption that the production rate of this nutrient is much faster than the motion of the cells, we can assume that its distribution is ruled by a statical PDE with no explicit time-dependence, and gives rise to a system which is a gradient flow in the variable $\rho$ (the parabolic-parabolic case, where the time scale for the cells and for the nutrient are comparable, is also a gradient flow, in the product space W2 x L 2 , but we will not consider this case). Since we mainly concentrate on the case of bounded domains, in the Keller-Segel case the term u[$\rho$] cannot be expressed as a convoluton and requires ad-hoc computations. In all the paper, the estimates will be studied on a time-discretized version of these PDEs, consisting in the so-called JKO (Jordan-Kinderleherer-Otto) scheme, based on iterated optimization problems involving the Wasserstein distance W2. We will first present 0-order estimates, on the L p and L $\infty$ norms of the solution. This is just a translation into the JKO language of well-known properties of these equations. The main goal of this part is hence to