# On the solutions of the variational problem $\min_\rho W_2^2(\rho,\nu)+F(\rho)$: new estimates and applications

## Filippo Santambrogio

### (Laboratoire de Mathématiques, Université Paris-Sud)

Date: Oct 30, 2014, time: 14:30

Abstract. The problem of minimizing the sum of the squared Wasserstein distance to a given measure plus a penalization appears very often in the applications of transport theory: it appears in the time-discretization of gradient flows or other evolution equations, in regularization procedures in image processing, in spatial economics...
I will review the optimality conditions for this problem and the consequences that they have in terms of estimates on the optimal $?$.
In particular I will concentrate on $L^\infty$ and $BV$ estimates in the case where $F$ is the integral of a convex function $f(?(x))$.
The BV case has been recently obtained with G. De Philippis, A. Mészáros and B. Velichkov, and has interesting applications to the problem of the Wasserstein projection onto the set of densities bounded by a given constant, that we constantly used with B. Maury in the optimal transport approach to crowd motion.
The $L^\infty$ case is more classical and mainly based on the use of the Monge-Ampère equation. Yet, a similar strategy can also be applied when F is the squared $H^{-1}$ norm (given by a logarithmic interaction term in dimension 2). This allows to deal with the parabolic-elliptic Keller-Segel equation and obtain very general
$L^\infty$ bounds for small time, under any type of diffusion, and for supercritical or subcritical mass. This part of the talk, where I will only sketch the ideas of the estimate, comes from a work-in-progress with J.-A. Carrillo.
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