Together with Antonio Esposito and Georg Heinze, we study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively.… Read the rest

# Author Archives: André

# Preprint: Variational convergence of the Scharfetter-Gummel scheme

Together with Anastasiia Hraivoronska and Oliver Tse, we explore the convergence of the Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite volume scheme that works consistently for any nonnegative diffusion constant, which allows us to study the discrete-to-continuum and zero-diffusion limits simultaneously.… Read the rest

# Published: On a class of nonlocal continuity equations on graphs:

The article with Antonio Esposito and Francesco Saverio Patacchini on *a class of nonlocal continuity equations on graphs* got published in the European Journal of Applied Mathematics. This is a follow-up work on our previous work also with Dejan Slepcev, where we introduced evolutions on graphs based on Upwind interpolation.… Read the rest

# Published: Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients

The paper with Víctor Navarro-Fernández on Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients got published at ESAIM: Mathematical Modelling and Numerical Analysis (M2AN). In the revision (also on arXiv:2201.10411), we arrived at uniform errors estimate in the diffusion constant also in the limit of vanishing diffusion.… Read the rest

# Preprint: Covariance-modulated optimal transport and gradient flows

Together with Martin Burger, Franca Hoffmann, Daniel Matthes and Matthias Erbar, we investigate a new dynamical optimal transport distance in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems.… Read the rest

# Special issue at EJAM: Evolution Equations on Graphs: Analysis and Applications

Together with Katy Craig (*UC Santa Barbara), *Antonio Esposito (*University of Oxford) and *Dimitrios Giannakis (*Dartmouth College),* we invite for contributions to the special issue **Evolution Equations on Graphs: Analysis and Applications** at European Journal of Applied Mathematics.… Read the rest

# Registration open and call for contributions at workshop on model structures for non-equilibrium systems, April 2023

We open the call to register and contribute talks/posters at the workshop

“In search of model structures for non-equilibrium systems”,

which will take place at the University of Münster, April 2023 – 28. April 2023.

Details: https://uni-muenster.de/MathematicsMuenster/go/non-equilibrium-systems

Please register using the link (the deadline for participation is March 27):

https://wwuindico.uni-muenster.de/event/1669/registrations/

# CIRM Workshop proposal accepted

The workshop proposal together with Pierre Monmarché, Julien Reygner and Marielle Simon on *PDE & Probability in interaction: functional inequalities, optimal transport and particle systems* got accepted. I’m happy to see you as speaker and participant from 22-26 January 2024 at CIRM.… Read the rest

# Published: Optimal stability estimates and a new uniqueness result for advection-diffusion equations

The paper with Víctor Navarro-Fernández and Christian Seis on *Optimal stability estimates and a new uniqueness result for advection-diffusion equations* got published at Pure and Applied Analysis.

# Pressure as Lagrange multiplicator in Stokes and Euler equation

\(\newcommand{\R}{\mathbb{R}}\)

[stextbox id=”Theorem” caption=”Navier-Stokes equation in \(d=2,3\)”]

\[ \begin{aligned}

u &: [0,\infty) \times \Omega \to \R^d &\text{velocity (unknown)} & \text{ length / time } \\

p &: [0,\infty) \times \Omega \to \R^d &\text{pressure (unknown)} & \text{ force / area } \\

f &: [0,\infty) \times \Omega \to \R^d &\text{external volume force (data)} & \text{ force / volume } \\

\varrho &\in [0,\infty) &\text{mass density (data)} & \text{ mass / volume } \\

\mu&\in (0,\infty) &\text{dynamic viscosity (data)} & \text{ pressure \(\cdot\) time }

\end{aligned}\]

\[\begin{aligned}

\varrho \left( \partial_t u + u \cdot \nabla u \right) – \mu \Delta u + \nabla p &= f &&\text{in \((0,\infty)\times \Omega\)},\\

\nabla \cdot u &= 0 &&\text{in \((0,\infty)\times \Omega\)},\\

u &= 0 &&\text{on \((0,\infty)\times \partial \Omega\)}

\end{aligned}\]

and initial conditions \(u(t=0,\cdot)=u_0(\cdot)\).… Read the rest