# Analysis, Geometry, and Dynamical Systems Seminar

## Next session

- 03/02/2015, 15:00 — 16:00 — Room P3.10, Mathematics Building

Léonard Monsaingeon,*CAMGSD* -
### A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations

In this talk I will discuss some recent results obtained with D. Kinderlehrer (Carnegie Mellon Univ.) and X. Xu (Purdue Univ.) for the Poisson-Nernst-Planck equations

\begin{equation}

t\geq 0,\,x\in {\mathbb R}^d,\,d\geq 3:\qquad

\begin{cases}

\partial_t u=\Delta u^m +\operatorname{div}(u\nabla (U+\Psi)),\\

\partial_t v=\Delta v^m +\operatorname{div}(v\nabla (V-\Psi)),\\

-\Delta \Psi =u-v.

\end{cases}

\label{eq:PNP}

\end{equation}The unknowns $u,v\geq 0$ represent the density of some positively and negatively charged particles, $U,V$ are prescribed confining potentials, $\Psi=(-\Delta)^{-1}(u-v)$ is the (nonlocal) self-induced electrostatic potential, and $m\geq 1$ a fixed non-linear diffusion exponent. We show that \eqref{eq:PNP} is the gradient flow of a certain energy functional in the metric space $\left(\mathcal{P}({\mathbb R}^d),\mathcal{W}_2\right)$ of Borel probability measures endowed with the quadratic Wasserstein-Rubinstein-Kantorovich distance $\mathcal{W}_2$. The gradient flow approach in $\left(\mathcal{P}({\mathbb R}^d),\mathcal{W}_2\right)$ is closely related to the theory of optimal mass transport and has successfully been employed for several scalar PDEs (Fokker-Planck, Porous Media, Keller-Segel, thin film...) We exploit this variational structure in order to semi-discretize (in time) the system and construct approximate solutions by means of the DeGiorgi minimizing movement. Sending the time step $h\downarrow 0$ we retrieve global weak solutions for initial data with low integrability and without regularity assumptions. In addition to energy monotonicity we also recover some regularity and new $L^p$ estimates. The proof deals with linear and nonlinear diffusions ($m=1$ and $m>1$) in a unified energetic framework.