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

# Author Archives: André

# 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

# weak L¹-convergence

\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)

Let \(\Omega\subset \mathbb{R}^n\), a sequence \((u_n)\) converges weakly to \(u\in L^p(\Omega)\) if

\[ \int_\Omega u_n v\;\dx{x} = \int_\Omega u v \;\dx{x} , \qquad \forall v\in L^\infty(\Omega) . \] As usual the convergence is denoted by \(u_n \rightharpoonup u\) in \(L^1(\Omega)\).

# Classic results for porous medium equation

### Selfsimilar solutions

\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)

Let us seek for solutions of \(\partial \varrho = \Delta \varrho^m\) satisfying the scaling hypothesis

\[

(x,t)\mapsto (y,\tau):=\left(\frac{x}{s(t)},\tau(t)\right) \qquad \varrho(x,t) = \frac{1}{s(t)^N} u\left(\frac{x}{s(t)},\tau(t)\right),

\]

where \(s: \R_+ \to \R_+\) and \(\tau:\R_+ \to \R_+\) are reparametrization of time and \(u:\R_+\times \R^N \to \R\).… Read the rest

# Fokker Planck equation

### Description

\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)

The Fokker Planck equation has the form

\[

\partial_t \varrho(x,t) = \nabla \cdot \bigl(\beta^{-1} \nabla \varrho(x,t) + \varrho(x,t) \nabla H(x)\bigr) ,\qquad \varrho(x,0)=\varrho_0(x) ,

\]

where \(H:\R^n\to \R\) is a smooth function, \(\beta>0\) some parameter and \(\varrho_0\) a probability density on \(\R^n\).… Read the rest