*Hans Humenberger (Univ. Wien (Austria))*

#### Abstract

Das wirklich gerechte Teilen einer Pizza ist – genau genommen – gar nicht so einfach. Mit einem geraden Messer muss man dabei immerhin den Mittelpunkt treffen, so dass der Schnitt ein Durchmesser ist. Dazu und auch für die Teilung unter mehreren Personen gibt es aber auch interessante Alternativen, die starken Bezug zur Elementargeometrie (Sekundarstufe 1) und auch zur Integralrechnung (Sekundarstufe 2) aufweisen. Im Vortrag sollen diese Alternativen vorgestellt und ihr Potential für die Lehre (Lehramtsstudium, Schule) besprochen werden. Darüber hinaus gibt es ein interessantes dreidimensionales Analogon für Kugeln („Äpfel“; die Kernidee dabei ist das wichtige und bekannte Prinzip von Cavalieri), das ebenfalls im Vortrag vorgestellt wird.

*Marius Müller (Universität Ulm (Germany))*

#### Abstract

In this talk, we examine steepest energy descent flows subject to obstacle constraints. We are interested in higher order frameworks, where the maximum principle does not apply. The constructed flows can be understood as gradient flows in the sense of Ambrosio, Gigli and Savare. To ensure existence of such flows, one often imposes continuity conditions on the metric slope of the energy, which turn out to be restrictive for nonlinear obstacle problems.

One can circumvent these conditions given a suitable compactness of the discrete trajectories in the De Giorgi Minimizing Movement Scheme. The required compactness is easy to obtain for a large class of fourth order equations with Navier boundary conditions.

An important application is the 1-dimensional Willmore flow with obstacle constraint, for which we will study long-time-behavior, finding asymptotic properties that match up with the results of A. Dall’Acqua and K. Deckelnick on the time-independent variational problem.

*Mathieu Lewin (CNRS/Paris-Dauphine (France))*

#### Abstract

The crystallization conjecture is one of the least understood problems in mathematical physics, and it is yet important for physical applications. In this talk I will explain what it is about on a simple class of models, sometimes called “Coulomb” or “Riesz” gases. This model is believed to be universal in a certain sense, and it has important implications in analytic number theory, in PDE, for random matrices, for random Schrödinger operators and several other mathematical topics. The talk will not contain any personal result.

*Michael Herty (RWTH Aachen University (Germany))*

*Peter Hornung (TU Dresden (Germany))*

*François Alouges (Ecole Polytechnique Paris (France))*

#### Abstract

Like any locomotion problem, driving micro mechanisms that can swim can be studied using tools of control theory. Nevertheless, this problem possesses some peculiarities that are consequences of the negligible inertia at that scale, the most famous being probably the obstruction formalized by E. M. Purcell as the “scallop theorem”. We will show how Lie brackets computations unveils the set of attainable directions that the swimmer may take, permitting to circumvent the obstruction. When energy minimizing conditions are furthermore required, as for optimizing Lighthill’s efficiency, the problem becomes an optimal control problem and optimal gaits can be geometrically expressed as suitable geodesics in the space of shape deformations. Finding periodic optimal gaits is a problem which is only tractable under numerical studies, and very little intuition can be extracted from the optimal strategies that are found in this way. However, if one considers furthermore the small amplitude stroke regime, it is possible to characterize optimal strokes in a way that produces naturally common features that are well known such as the bending wave along Taylor sheet. Other Mechanisms such as a swimmer with 3 arms that can be elongated will be also studied. Results shown in the talk are joint works with A. DeSimone, G. DiFratta, L. Giraldi, Y. Or, O. Wiezel and M. Zoppello.

*Michael Herty (RWTH Aachen University (Germany))*

*Rüdiger Schultz (Univ. Duisburg-Essen (Germany))*

#### Abstract

The newsvendor is among the classic problems in decision making under uncertainty. It consists of having to purchase for later use a commodity under uncertain demand. At demand realization, excess and shortfall of the commodity induce compensation costs; that add to the initial purchase cost. The task is to minimize total cost. In the talk we quickly depart from the newsvendor, extract basic features of two-stage stochastic programming, and visit more recent classes of stochastic programs. The first is PDE constrained optimization, where optimization of shapes under linearized elasticity and random loading is treated in detail. The second class of problems lives in finite-dimension. It is two-stage stochastic mixed-integer linear programs. For both classes we discuss structural properties beyond convexity, elaborate on solution methods, and report numerical testing for practical and academic applications.

*Michael Westdickenberg (RWTH Aachen University (Germany))*

#### Abstract

We discuss several aspects related to energy dissipation in incompressible fluid flows: model derivation, Leray’s existence result, energy cascades, turbulent flows, anomalous dissipation, Onsager conjecture, nonuniqueness for incompressible Euler.

*Michael Westdickenberg (RWTH Aachen University (Germany))*

#### Abstract

We discuss several aspects related to energy dissipation in incompressible fluid flows: model derivation, Leray’s existence result, energy cascades, turbulent flows, anomalous dissipation, Onsager conjecture, nonuniqueness for incompressible Euler.

*Fritz Hiesmayr (University of Cambridge (UK))*

#### Abstract

In the Allen-Cahn construction one constructs minimal surfaces as the weak limit of level sets of a semilinear elliptic PDE, called the Allen-Cahn equation. In this talk I will give a brief overview of the Allen-Cahn theory, and then present my work relating the variational properties of the solutions of the Allen-Cahn equation to the surface arising in the limit. For example, the minimal surface inherits index bounds from a sequence of unstable critical points; time permitting I will sketch a proof of this.

*Heiko von der Mosel (RWTH Aachen University (Germany))*

#### Abstract

The basic notion to distinguish different knot types is that of an ambient isotopy. In this introductory talk we discuss various conditions ensuring that two given knots are of the same type, and present a few applications in geometric knot theory.

*Maria J. Esteban (CEREMADE Université Paris Dauphine (France))*

#### Abstract

In this talk I will present how to prove interpolation inequalities, with as good constants as possible, for Schrödinger operators involving an external magnetic field. These inequalities can be then used to prove spectral estimates for the same kind of operators, but this time involving also an external electrical field.

Qualitative properties of the extremal functions in the inequalities will be also discussed.

The work presented in this talk has been obtained in collaboration with J. Dolbeault, A. Laptev and M. Loss.

*Sebstian Noelle (RWTH Aachen University (Germany))*

#### Abstract

A key issue of our RTG is entropy dissipation. For scalar conservation laws, there is a rich set of entropy pairs, and the corresponding entropy inequalities assure uniqueness. It is technically more difficult to prove the discrete analogues of entropy inequalities for numerical schemes, and indeed it is only possible for a rather limited class of schemes. I will review this and briefly comment how the entropy inequalities assure convergence, uniqueness and error estimates. This part of the talk is based on work of Oleinik (1950’s), Kruzkov (1970), Lax (1975), Cockburn (1987), Kröner (1995) and many others.

For isentropic flows, the (energy, energy-flux) pair has the mathematical structure of an entropy pair. The corresponding inequality does not assure uniqueness of weak solution. Still, a discrete energy inequality is considered a key estimate to assure stability of numerical schemes. Here we follow a framework proposed by Bouchut (2003).

*Michael Herty (RWTH Aachen University (Germany))*

#### Abstract

We are interested in feedback control design for interacting particle systems. The controlled system should be stable, independent of the number of particles and the computational costs is suffciently small. Further, we aim to derive feedback control consistent with the mean.

*Michael Westdickenberg (RWTH Aachen University (Germany))*

#### Abstract

We consider a hybrid compressible/incompressible system with memory effects, introduced recently by Lefebvre-Lepot and Maury for the description of one-dimensional granular flows. We prove a global existence result for this system without assuming additional viscous dissipation. Our approach extends the one by Cavalletti et al. for the pressureless Euler system to the constrained granular case with memory effects. We construct Lagrangian solutions based on an explicit formula using the monotone rearrangement associated to the density. We explain how the memory effects are linked to the external constraints imposed on the flow.

**Time**

September 4-5, 2018 (Tue/Wed)

**Location**

Landhotel Kallbach

Simonskall 24-26, 52393 Hürtgenwald

*This event is for members of the RTG only.*

## Program

### Tuesday, September 4

9:30-10:00h**Opening Remarks**

Michael Westdickenberg

10:00-10:45h**Status Report + Discussion***Why optimal transport of differential forms is cool*

Marco Doemeland

Coffee Break

11:15-12:00h**Status Report + Discussion***Well-balanced numerical schemes for hyperbolic balance laws in networks*

Yogiraj Mantri

12:00-12:30h**Formal Reviews**

*Sarah Biesenbach*, Michael Herty, Michael Westdickenberg*Marco Doemeland*, Sebastian Noelle, Manuel Torrilhon*Nikolas Siccha*, Christof Melcher, Heiko von der Mosel

Lunch

14:00-15:00h

Research Seminar

Michael Westdickenberg

RWTH Aachen University (Germany)**One-dimensional granular system with memory effects**

Abstract

15:00-15:45h**Status Report + Discussion***How to derive a metastability result of solutions to the Cahn-Hilliard equation*

Sarah Biesenbach

Coffee Break

16:15-17:00h**Status Report + Discussion***Gradient flow for the Menger curvature energy of measures*

Hannes Matt

17:00-18:00h**Working Session**

*Sarah Biesenbach*, Heiko von der Mosel*Marco Doemeland*, Christof Melcher, Michael Westdickenberg*Yogiraj Mantri*, Michael Herty, Sebastian Noelle

18:00-19:00h**Working Session**

*Hannes Matt*, Heiko von der Mosel, Michael Westdickenberg*Yannick Holle*, Michael Herty

Dinner

20:00-21:00h**Show + Tell**

### Wednesday, September 5

9:00-10:30h**Strategy Session**

Plenum

Coffee Break

11:00-12:00h**Planning Session**

Steering Committee

12:00-12:30h**Formal Reviews**

*Wadim Gerner*, Michael Herty, Manuel Torrilhon*Yogiraj Mantri*, Maria Westdickenberg, Michael Westdickenberg*Hannes Matt*, Christof Melcher, Sebastian Noelle

Lunch

14:00-15:00h

Research Seminar

Michael Herty

RWTH Aachen University (Germany)**Multi-agent systems and games**

Abstract

15:00-15:45h**Status Report + Discussion***Existence of magnetic energy minimisers of constant helicity*

Wadim Gerner

Coffee Break

16:15-17:00h**Status Report + Discussion***GENERIC, hyperbolic PDEs and geometric integrators*

Nikolas Siccha

17:00-18:00h**Working Session**

*Wadim Gerner*, Christof Melcher, Heiko von der Mosel*Nikolas Siccha*, Manuel Torrilhon, Michael Westdickenberg

*Tim Laux (UC Berkeley (USA))*

#### Abstract

In this talk I will present a recent work with Guido de Philippis on the Almgren-Taylor-Wang time discretization for mean curvature flow in the case of outward minimizing sets. I will show that the scheme preserves outward minimality and, by compensated compactness techniques, that the arrival time functions converge strictly in BV. In particular, this establishes the convergence of the time-integrated perimeters of the approximations. As a corollary, the conditional convergence result of Luckhaus-Sturzenhecker becomes unconditional in the outward minimizing case. Most parts of the talk should be accessible for Master’s students with some background in geometric analysis and/or calculus of variations.

*Wadim Gerner (RWTH Aachen University (Germany))*

#### Abstract

In ideal hydrodynamics a fluids motion is characterised by the Euler equations. We will concern ourselves with time-independent solutions of the Euler equations, which give rise to steady flows. In particular we will analyse the topological properties of the integral curves of solutions of the incompressible stationary Euler equations on compact, oriented, Riemannian $3$-manifolds. The aim of this talk will be to present the description theorem for real analytic steady flows [AK98] p.69.

The theorem roughly states that if a corresponding solution of the Euler equations is not everywhere collinear with its curl, our manifold is divided into finitely many cells, each of which decomposes into surfaces which are diffeomorphic to the torus and invariant under the flow of the fluid.

In the first part of this talk I will shortly review some of the concepts of differential geometry which are needed to formulate and understand the problem at hand. I will then proceed to discuss the case of general differentiable flows before turning to the real analytic case. Lastly, if time allows, I want to present how solutions of the stationary incompressible Euler equations may be viewed as solutions of the Euler-Lagrange equation of an appropriate energy functional and hence provide a variational view on steady flows.

#### References

[AK98]

V. I. Arnold and B. A. Khesin, *Topological Methods in Hydrodynamics* (1998),
Springer Verlag

*Wadim Gerner (RWTH Aachen University (Germany))*

#### Abstract

In ideal hydrodynamics a fluids motion is characterised by the Euler equations. We will concern ourselves with time-independent solutions of the Euler equations, which give rise to steady flows. In particular we will analyse the topological properties of the integral curves of solutions of the incompressible stationary Euler equations on compact, oriented, Riemannian $3$-manifolds. The aim of this talk will be to present the description theorem for real analytic steady flows [AK98] p.69.

The theorem roughly states that if a corresponding solution of the Euler equations is not everywhere collinear with its curl, our manifold is divided into finitely many cells, each of which decomposes into surfaces which are diffeomorphic to the torus and invariant under the flow of the fluid.

In the first part of this talk I will shortly review some of the concepts of differential geometry which are needed to formulate and understand the problem at hand. I will then proceed to discuss the case of general differentiable flows before turning to the real analytic case. Lastly, if time allows, I want to present how solutions of the stationary incompressible Euler equations may be viewed as solutions of the Euler-Lagrange equation of an appropriate energy functional and hence provide a variational view on steady flows.

#### References

[AK98]

V. I. Arnold and B. A. Khesin, *Topological Methods in Hydrodynamics* (1998),
Springer Verlag

*Simon Foucart (Texas A&M University (U.S.A.))*

#### Abstract

This talk aims at advocating interplays between Optimization and Approximation Theory by describing two successful examples. In the first example, we exploit a semidefinite characterization for the nonnegativity of univariate polynomials to devise a method for computing best approximants by splines under constraints. This is implemented as a MATLAB package called Basc. In the second example, we leverage the method of moments to tackle the long-standing problem of minimal projections. In particular, we determine numerically the projection constants for the spaces of cubic, quartic, and quintic univariate polynomials.

*Dario Trevisan (University of Pisa (Italy))*

#### Abstract

We illustrate a PDE method to prove asymptotic rates of transportation costs of random empirical measures. Our technique is based on a rigorous “linearization” of the Monge-Ampère equation proposed in the physical literature. As a main application, we obtain for the first time the leading term in the expansion of the 2-Wasserstein distance for independent uniformly distributed points in a square. We will also mention more recent developments and open problems, such as convergence of the transportation maps, and variations in the geometry of the space or the correlations of points. (Joint work with L. Ambrosio and F. Stra)

*Manuel Torrilhon (RWTH Aachen University (Germany))*

*Michael Herty (RWTH Aachen University (Germany))*

*Sarah Biesenbach (RWTH Aachen University (Germany))*

#### Abstract

This talk is concerned with the time-dependent metastable behaviour of solutions to the scalar Cahn-Hilliard equation on the compact interval $\left[ -\frac{\Lambda}{2}, \frac{\Lambda}{2} \right]$, subject to periodic boundary conditions.

The Cahn-Hilliard equation is the gradient flow of the Ginzburg-Landau energy with respect to the $\dot{H}^{-1}$ metric. In general, local minimizers of an energy are the stable sets of the corresponding gradient flow. Solutions started at the minimizers are in equilibrium and solutions started nearby get drawn towards the equilibrium state.

Solutions of the Cahn-Hilliard equation exhibit metastability. They relax towards a metastable set first. This so-called $N$-layer branch of the slow manifold is constructed by piecing together local energy minimizers. It contains functions $v$ with large phases where $v = \pm 1$ and with $N$ transition layers connecting the $\pm 1$ phases.

Solutions with initial data that is order one away from the slow manifold are drawn rather quickly towards the metastable set (this phase beeing called relaxation) where they remain trapped for an extremely long time. The solution appears to be stable. However, once that extremely long time is over, the two closest layers come together, collide and annihilate, leaving behind a solution that is order one away from the $N-2$ layer metastable set. The process of fast relaxation towards the slow manifold, the metastable pattern and the collision of the two closest of the remaining layers repeats until there are only two layers left. The problem may therefore be labelled as a coarsening problem.

During a collision there is excess energy in the dynamic system. Hence, in order to track the evolution through layer collisions, one needs to control initial data that is order one away from the slow manifold. This has been done in [SW].

During my talk I will focus on the dynamic metastability result in [SW] in which the authors establish metastable behaviour for a broad class of initial data. I will briefly comment on earlier works in the literature where similar results are obtained, but only for well-behaved initial data. I will then comment on the ideas and methods presented in [SW]. We will see that, in order to capture the initial relaxation for data started order one away from the metastable set, it is necessary to understand the algebraic-in-time relaxation of the problem on the real line.

#### References

[SW]
S. Scholtes and M. G. Westdickenberg,
*Metastability of the Cahn-Hilliard equation in one space dimension*,
J. Differential Equations **265** (2018), 1528-1575

*Marco Doemeland (RWTH Aachen University (Germany))*

#### Abstract

In this talk we investigate a strong analogy between Mather’s theory of minimal measures in Lagrangian dynamics and the mass transportation (or Monge-Kantorovich) problem. First noticed by L. C. Evans, this analogy was formulated by De Pascale et al. [DGG2006] with the help of a minimization problem for closed $1$-currents. In a special case the reformulation of Mather’s problem with $1$-currents was already done by Bangert [B1999] by interpreting a minimal measure on the tangent bundle as a mass minimizing $1$-current. The reformulation of the mass transportation problem with $1$-currents is done with the help of weak solutions of the continuity equation.

The talk will be split into two parts. In the first part I will give a short introduction on the notion of currents and state the equivalent problems. In the second part we dive a little bit into the ideas behind the proof of the equivalences.

#### References

[B1999]

V. Bangert, *Minimal Measures and Minimizing Closed Normal One-currents*, Geometric & Functional Analysis **9** (1999), no. 3, 413-427.

[DGG2006]

L. De Pascale, M. S. Gelli, and L. Granieri, *Minimal measures, one-dimensional currents and the Monge-Kantorovich problem*, Calculus of Variations and Partial Differential Equations **27** (2006), no. 1, 1-23.

*Marco Doemeland (RWTH Aachen University (Germany))*

#### Abstract

In this talk we investigate a strong analogy between Mather’s theory of minimal measures in Lagrangian dynamics and the mass transportation (or Monge-Kantorovich) problem. First noticed by L. C. Evans, this analogy was formulated by De Pascale et al. [DGG2006] with the help of a minimization problem for closed $1$-currents. In a special case the reformulation of Mather’s problem with $1$-currents was already done by Bangert [B1999] by interpreting a minimal measure on the tangent bundle as a mass minimizing $1$-current. The reformulation of the mass transportation problem with $1$-currents is done with the help of weak solutions of the continuity equation.

The talk will be split into two parts. In the first part I will give a short introduction on the notion of currents and state the equivalent problems. In the second part we dive a little bit into the ideas behind the proof of the equivalences.

#### References

[B1999]

V. Bangert, *Minimal Measures and Minimizing Closed Normal One-currents*, Geometric & Functional Analysis **9** (1999), no. 3, 413-427.

[DGG2006]

L. De Pascale, M. S. Gelli, and L. Granieri, *Minimal measures, one-dimensional currents and the Monge-Kantorovich problem*, Calculus of Variations and Partial Differential Equations **27** (2006), no. 1, 1-23.

*Henrik Schumacher (RWTH Aachen University (Germany))*

#### Abstract

The highlight of this talk (and the whole course) will be the construction of a complete metric on the shape space of unparameterized immersions. To this end, we have to review the quotient construction - which happens to be somewhat broken in the category of metric spaces. If time allows, I’ll also try to outline how this metric relates to other popular shape metrics such as Hausdorff distance, Fréchet distance, and Gromov-Hausdorff distance.

*Paolo Bonicatto (University of Basel (Switzerland))*

#### Abstract

Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation

$$ \partial_t u + \mathbf b \cdot \nabla u=0 $$

where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.

In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $\mathrm{BV}$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $\mathrm{BV}$ vector fields and thus gives a positive answer to the (weak) Bressan’s Compactness Conjecture. This is a joint work with S. Bianchini.

*Henrik Schumacher (RWTH Aachen University (Germany))*

#### Abstract

In diesem Vortrag möchte ich vorführen, wie man den Raum der Lipschitz-Immersionen

$$ f \colon \varsigma \to \mathbb{R}^m $$

so metrisieren kann, dass man einen vollständigen metrischen Raum erhält. Wesentlicher Baustein für diese Metrik wird eine $L^\infty$-artige Metrik auf dem Raum der Riemannschen Metriken der Klasse $L^\infty$ auf $\varsigma$.

Hierzu diskutiere ich zunächst eine Familie von vollständigen Riemannschen Metriken auf dem Raum der positiv-definiten, symmetrischen Bilinearformen auf einem endlichdimensionalen, reellen Vektorraum $V$. Diese Riemannschen Metriken sind dadurch charakterisiert dass sie invariant sind unter Zurückziehung entlang von Elementen $A \in \mathrm{GL}(V)$.

Diese Familie ist auch wesentlicher Bestandteil von Henckys Modell der elastischen Verformungsenergie eines isotropen Mediums; dieses Modell wird vielfach auch „logarithmic strain model“ oder „true strain model“ genannt. Diesen Zusammenhang möchte ich bei der Gelegenheit auch umreißen, insbesondere, da dies an dieser Stelle (für elastizitätstheoretische Begrifflichkeiten) relativ leicht zugänglich ist und auch eine gewisse Intuition liefert.

In der darauffolgenden Sitzung möchte ich dann diese Konstruktion verwenden, um den shape space der Lipschitz-Immersionen (d.h. den Raum der *unparametrisierten* Lipschitz-Immersionen) mit einer vollständigen Metrik zu versehen.

*Giuseppe Visconti (RWTH Aachen University (Germany))*

#### Abstract

In this talk we will present some preliminary results on kinetic models for big data. These models are the multi-dimensional generalization of microscopic models for opinion dynamics. They are characterized by the property of being able to partition a large set of data in clusters. The kinetic limit allows us to easily investigate the properties of the model and to easily treat a large amount of data. Applications to pattern recognition problems are presented.

*Torsten Trimborn (RWTH Aachen University (Germany))*

#### Abstract

We introduce a large system of interacting financial agents in which each financial agent is faced with the decision of how to allocate his capital between a risky stock or a risk-less bond. The investment decision of investors, derived through an optimization drives the stock price. The model has been inspired by the econophysical Levy-Levy-Solomon model [1]. We derive the mesoscopic description of our model in order to analyze the wealth and stock price distribution. We focus on the behavioral aspects in our model and discuss the impact on the distribution functions.

[1] H. Levy, M. Levy, S. Solomon, Microscopic simulation of financial markets: from investor behavior to market phenomena, Elsevier, 2000.

*Martin Frank (Karlsruhe Institute of Technology (Germany))*

#### Abstract

This talk considers the derivation of the mean field game equations from a specific control problem for a system of ODEs, in the limit of infinitely many players. We will have a look at non-technical and seemingly technical assumptions underlying the limit. We especially study scaling assumptions on parameters.

*Giacomo Albi (Università degli Studi di Verona (Italy))*

#### Abstract

In this talk, we will start revising some of the main results on the optimal control of interacting agent systems. In particular, we will focus on consensus/aggregation dynamics, which have risen a lot of interest in many applications such as cell motion, opinion formation, and pedestrian dynamics. For large systems of agents a mean field description will be introduced, consisting in optimal control problems constrained by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population.

In order to numerically solve these type of problems, we will firstly propose a class of approximating hierarchy of sub-optimal controls based on a Boltzmann-Bellmann approach, whose computation requires a very moderate numerical complexity with respect to direct approaches. In a second step, we will tackle directly the optimal control problem of Boltzmann-type equations, providing a consistency results in relation to the solution of mean-field optimal control problems.

Numerical experiments will validate the theoretical findings, comparing different strategies, and settings in a consensus modelling framework.

*Mattia Zanella (Politecnico di Torino (Italy))*

#### Abstract

Kinetic equations play a major role in the modelling of large systems of interacting particles/agents with a proved effectiveness in describing real world phenomena ranging from plasma physics to socioeconomic dynamics. Their formulation has often to deal with physical, or even “social”, forces deduced empirically and of which we have at most statistical information. Hence, to produce realistic descriptions of the underlying phenomena it is of paramount importance to consider the presence of random inputs in the form of uncertain parameters as a structural feature of the kinetic models and to develop suitable numerical methods to capture admissible states of the systems [2,4].

In this talk we concentrate on stochastic Galerkin methods for the uncertainty quantification of Vlasov-Fokker-Planck (VFP) equations with nonlocal flux. In particular, we develop methods that preserve their large-time solution [2,3] and we introduce the so-called Monte Carlo generalized polynomial chaos (MCgPC) methods [1]. In contrast to a direct application of stochastic Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without losing nonnegativity of the solution.

[1] J. A. Carrillo, L. Pareschi, M. Zanella. Particle based gPC methods for mean-field models of swarming with uncertainty. Communications in Computational Physics, to appear.

Preprint arxiv:1712.01677.

[2] G. Dimarco, L. Pareschi, M. Zanella. Uncertainty quantification for kinetic models in socio-economic and life sciences. In Uncertainty Quantification for Hyperbolic and Kinetic Equations, Editors S. Jin, and L. Pareschi, SEMA SIMAI Springer Series, vol. 14, 2017.

[3] G. Dimarco, L. Pareschi, M. Zanella. Micro-macro generalized polynomial chaos techniques for kinetic equations. Work in progress.

[4] A. Tosin, M. Zanella. Boltzmann-type models with uncertain binary interactions. Communications in Mathematical Sciences, to appear.

Preprint arxiv:1709.02353.

*Lorenzo Pareschi (Università degli Studi di Ferrara (Italy))*

#### Abstract

In this talk we will present some recent results on the construction of efficient Monte Carlo methods for uncertainty quantification in kinetic equations. The methods are based on a suitable micro-macro decomposition strategy combined with the use of different sampling scales on the random field in order to reduce the variance of standard Monte Carlo sampling methods. Applications to socio-economy and rarefied gas dynamics are presented.

Control problems and differential games for agent based systems have been recently studied with methods of kinetic theory. In particular, under the presence of uncertainty kinetic theory provides novel techniques to treat problems of high dimensionality and complexity. The workshop aims to bring together researchers in this field and will provide a great opportunity to initiate scientific collaboration amongst participants. The idea is to bring researchers to discuss recent advances in kinetic modelling and numerical methods with application to large-scale agent based systems.

- Time
- May 14-15, 2018 (Mon-Tue)
- Location
- RWTH Aachen University, Turmstraße 46, Room 301

To register for the event, please send an email to info@eddy.rwth-aachen.de.

There is no conference fee.

## Program

The workshop will take place in room 301 in Turmstraße 46, 52064 Aachen.

### Monday, May 14

14:00-14:45h

Lorenzo Pareschi

Università degli Studi di Ferrara (Italy)**Multiscale Monte Carlo methods for uncertainty quantification of kinetic equations**

Abstract

14:45-15:15h

Mattia Zanella

Politecnico di Torino (Italy)**Stochastic Galerkin methods for kinetic equations of the collective behavior**

Abstract

Coffee Break

15:45-16:30h

Giacomo Albi

Università degli Studi di Verona (Italy)**Boltzmann-type optimal control problems**

Abstract

16:30-18:00h

Work Session - Discussion

### Tuesday, May 15

9:00-9:45h

Martin Frank

Karlsruhe Institute of Technology (Germany)**Formal derivation of Mean Field Game Equations**

Abstract

9:45-10:15h

Torsten Trimborn

RWTH Aachen University (Germany)**A Behavioral Portfolio Model: A Kinetic Approach**

Abstract

Coffee Break

10:45-11:15h

Giuseppe Visconti

RWTH Aachen University (Germany)**Kinetic models for big data**

Abstract

11:15-12:00h

Giacomo Dimarco

Università degli Studi di Ferrara (Italy)**TBA**

Abstract

Lunch Break

14:00-18:00h

Work Session - Discussion

## Travel

### Accommodation

Hotel rooms in the center of Aachen can be booked through the online registration form available at the following link
(which takes you to the **aachen tourist service e.v.** website):

Upon booking you will receive a confirmation email. The rates listed there are per room and per day, and include taxes. Breakfast may or may not be included. There are no additional registration fees. The booked rooms will be available from the first night of the reservation.

### How to get to Aachen…

Aachen is well connected to the train system in Germany. Tickets can be purchased online at Deutsche Bahn. There are fast train connections to Paris, Brussels, and London operated by Thalys in combination with Eurostar.

- From Frankfurt Airport
- The train ride from Frankfurt to Aachen takes about two to three hours, depending on the connections, with a changeover at Cologne Main Train Station. It costs between 51 and 74 Euros, depending on if you take a regional train with stops in between or quicker ICE connection with the Deutsche Bahn. Tickets can be acquired in the train office at the airport with either cash or credit card.
- From Düsseldorf Airport
- Depending on the connections, the train ride from Düsseldorf to Aachen takes about one and a half to two hours with the Deutsche Bahn. Some connections require a changeover at Düseldorf Main Train Station or Cologne Main Train Station. Costs range from 18-28 Euros. Tickets can be bought in the train office at the airport with either cash or credit card.
- Fom Cologne/Bonn Airport
- Depending on the connections, the train ride with Deutsche Bahn
from Cologne/Bonn Airport to Aachen takes about one to one and half hours with a changeover at Cologne Main Train Station.
Costs range from 15-21 Euros.
Tickets can be bought in the train office at the airport with either cash or credit card.

## Organizer

Michael Herty

RWTH Aachen University (Germany)

## Sponsors

Deutscher Akademischer Austauschdienst (DAAD)

RWTH Aachen University

Deutsche Forschungsgemeinschaft (DFG)

Università degli Studi di Ferrara

*Henrik Schumacher (RWTH Aachen University (Germany))*

#### Abstract

In diesem Vortrag möchte ich vorführen, wie man den Raum der Lipschitz-Immersionen

$$ f \colon \varsigma \to \mathbb{R}^m $$

so metrisieren kann, dass man einen vollständigen metrischen Raum erhält. Wesentlicher Baustein für diese Metrik wird eine $L^\infty$-artige Metrik auf dem Raum der Riemannschen Metriken der Klasse $L^\infty$ auf $\varsigma$.

Hierzu diskutiere ich zunächst eine Familie von vollständigen Riemannschen Metriken auf dem Raum der positiv-definiten, symmetrischen Bilinearformen auf einem endlichdimensionalen, reellen Vektorraum $V$. Diese Riemannschen Metriken sind dadurch charakterisiert dass sie invariant sind unter Zurückziehung entlang von Elementen $A \in \mathrm{GL}(V)$.

Diese Familie ist auch wesentlicher Bestandteil von Henckys Modell der elastischen Verformungsenergie eines isotropen Mediums; dieses Modell wird vielfach auch „logarithmic strain model“ oder „true strain model“ genannt. Diesen Zusammenhang möchte ich bei der Gelegenheit auch umreißen, insbesondere, da dies an dieser Stelle (für elastizitätstheoretische Begrifflichkeiten) relativ leicht zugänglich ist und auch eine gewisse Intuition liefert.

In der darauffolgenden Sitzung möchte ich dann diese Konstruktion verwenden, um den shape space der Lipschitz-Immersionen (d.h. den Raum der *unparametrisierten* Lipschitz-Immersionen) mit einer vollständigen Metrik zu versehen.

*Michael Hinze (University of Hamburg (Germany))*

#### Abstract

With this talk we present a novel reduced-basis approach for optimal control
problems with constraints, which seems to deliver lower dimensional RB spaces as
reported in the literature so far for the same problem class, but with the same
approximation properties, and which allows to prove an error equivalence as
known from finite element a posteriori error analysis. Numerical test confirm
our theoretical findings.**Please note: different time!**

*Nikolas Siccha (RWTH Aachen University (Germany))*

#### Abstract

The “General Equation for the NonEquilibrium Reversible-Irreversible Coupling” (GENERIC) was introduced by Grmela and Oettinger in 1997 as a framework for the formulation of thermodynamically admissible time evolutions of nonequilibrium systems.

In the first part of this talk we will give a short overview of the main building blocks of GENERIC and connect them to their precursors, namely reversible kinematics formulated in terms of Poisson brackets and irreversible kinematics formulated in terms of Ginzburg-Landau brackets. We will present several examples of well-known ODEs and PDEs cast into the GENERIC structure, and briefly talk about parallels and differences in their two formulations.

Next we discuss geometric integrators in the context of ODEs and PDEs in GENERIC form. The success of symplectic integrators for Hamiltonian ODEs and symmetric integrators for reversible ODEs suggests that there is something to be gained in applying variants or generalizations of these geometric integrators to PDEs that fit into the GENERIC formalism. We will first present motivations for applying geometric integrators to appropriate ODEs and then present a novel reversible-irreversible time integrator based on the discontinuous Galerkin time stepping method and its adjoint method.

Finally we present results of numerical experiments, for which we apply said time integrator to various linear problems and compare its convergence and long-time behaviour with standard time integrators.

*Michael Westdickenberg (RWTH Aachen University (Germany))*

#### Abstract

Gradient flows are a special class of ordinary differential equations, whose solutions are characterized as those curves, along which an energy/entropy function is decreasing at maximal rate (steepest descent). We will discuss various useful properties of these systems and outline generalizations to infinite-dimensional Banach or even metric spaces.

*Enno Lenzmann (University of Basel (Switzerland))*

#### Abstract

The energy-critical half-wave maps equation arises as a classical continuum limit of Calogero-Moser and Haldane-Shastry type spin systems in one space dimension. In this talk, I will discuss some essential features such as the complete classification of traveling solitary waves with finite energy, by using a close connection to minimal surfaces with free and non-free boundary conditions. Furthermore, I will present a recently found Lax pair structure and we explain its potential applications to the dynamics of the half-wave maps equation. Finally, I will mention some open problems. This talk is based on joint work with P. Gérard (Orsay) and A. Schikorra (Pittsburgh).

*Michael Westdickenberg (RWTH Aachen University (Germany))*

#### Abstract

Gradient flows are a special class of ordinary differential equations, whose solutions are characterized as those curves, along which an energy/entropy function is decreasing at maximal rate (steepest descent). We will discuss various useful properties of these systems and outline generalizations to infinite-dimensional Banach or even metric spaces.

*Guoxian Chen (Wuhan University (China) & Université d'Orléans (France))*

#### Abstract

We design a scheme for the Euler equations with gravity by our new designed
subcell hydrostatic reconstruction method, which is originally for the shallow
water equations [*SIAM Journal on Numerical Analysis*, 55(2):758-784, 2017]. The
key difficulty for such problem is to give a proper definition of the
non-conservative product of measures due to the gravitational potential, such
that two properties are preserved: one is the well-balancing property, which
means that the scheme can preserve the isothermal steady state exactly; another
one is the positive preserving property, which is preserve the non-negativity of
both gas density and pressure. The key idea of the scheme is splitting
singularity to be an infinite thin layer, on where the potential is smoothed
and the isothermal state variables are extended. The numerical experiments
demonstrate the robustness of the scheme.

*Hannes Matt (RWTH Aachen University (Germany))*

#### Abstract

In this talk I present the so-called “David-Léger curvature theorem” and a proof given by Xavier Tolsa. The theorem states that subsets of the plane with finite length and finite Menger curvature energy can essentially be covered by a countable union of Lipschitz graphs.

The Menger curvature of three pairwise different points is obtained by squaring the inverse of the radius of the circle passing through these points. This curvature is a non-negative function of three variables, making it a suitable energy density for a triple integral energy functional. In particular, when integrated in all three variables with respect to the one-dimensional Hausdorff measure restricted to some set with finite length, this yields the Menger curvature energy of said set.

*Yogiraj Mantri (RWTH Aachen University (Germany))*

#### Abstract

Gas flow through pipeline network can be described using 2x2 hyperbolic balance law along with coupling conditions at the nodes of the network. The numerical solution at steady state is highly sensitive to the coupling conditions at the nodes and the balance between the flux and source terms within the pipes. To avoid spurious oscillations for near equilibrium flows, it is essential to design well-balanced schemes. Recently Chertock, Herty & Özcan (2017) introduced a well-balanced method for 2x2 systems of balance laws. In the present talk, we expand this approach to include coupling conditions and compressors for gas pipeline network. We also prove the well-balancing for single pipe-to-pipe connection. First numerical experiments demonstrate stability for T-junctions as well.

*Hamed Zakerzadeh (Université Toulouse III (France))*

#### Abstract

Our goal here is to provide a complete convergence analysis of the Implicit-Explicit Particle-In-Cell (IMEX-PIC) methods introduced in [Filbet and Rodrigues, 2016] for the Vlasov-Poisson system with a strong external magnetic field. This is a joint work with F. Filbet (Toulouse) and M. Rodrigues (Rennes).

*Manuel Torrilhon (RWTH Aachen University (Germany))*

#### Abstract

Gas flow in thermal non-equilibrium can be described by kinetic theory and Boltzmann equation at the expense of strongly increased computational costs. A hierarchical simulation approach for Boltzmann’s equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurate and efficient as in computational fluid dynamics (CFD), but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. The hierarchical nature of the method can be used to provide efficient model error estimates to increase the predictivity of a CFD computation. This talk will give a proof-of-concept approach for such a framework for the case of the steady linearized Boltzmann equation using Hermite discretization, that is, moment equations and an implicit discontinuous Galerkin formulation.

*Henning Struchtrup (University of Victoria (Canada))*

#### Abstract

The promise of the method of moments is to capture the main gas behavior as described by the Boltzmann equation, at lower computational cost. For this, the Boltzmann equation is replaced by a finite set of moment equations, which is closed by a well chosen particle distribution function. If the latter is obtained by the Maximum Entropy Closure, the resulting closed set of moment equations is fully equivalent with Extended Thermodynamics, and sports, e.g., a proper second law and a generalized Gibbs equation for non-equilibrium. While there is some recent success in applying the Maximum Entropy Closure, it is presently limited to small moment sets, for one-dimensional equations. The far more convenient Grad closure allows arbitrary moment sets, but looses thermodynamic structure, which can only be guaranteed for the linearized equations. Nevertheless, the non-linear moment equations give meaningful results, and can be useful tools. This, obviously, requires the choice of the proper moment set, and availability of wall boundary conditions.

To answer the question of which and how many moments are required, the Order of Magnitude Method starts out with an arbitrarily large set of moment equations, and then uses smallness parameters, in particular the Knudsen number, to reduce the system to its essential content. This not only reduces the number of moments and equations to the required minimum (based on the smallness parameters), but also constructs the actual moment basis functions, which depend on the molecular interaction potential of the gas. The resulting moment sets alternate between (mainly) hyperbolic equations, as in Extended Thermodynamics, and equations of mixed type with parabolic contributions. The latter actually behave better than the hyperbolic ones, in particular they allow for relatively easy access to boundary conditions.

Strictly, the reduction of the large moment set is valid only in the bulk, but not close to walls where the Knudsen boundary layers describe the transition between the bulk gas and the discontinuity imposed by the wall. While moment systems provide good approximation to Knudsen layers, there is no clear-cut mathematical background that relates moment choice and numbers to accuracy of the Knudsen layer approximation.

More complicated molecular models with two or more characteristic times pose additional difficulty. For example, polyatomic gases have (at least) two time scales, and their relative magnitude depends on the overall Knudsen number, which leads to different ordering of various contributions at different length scales.

All of the above will be discussed, and illustrated with simulations of interesting rarefied gas flows. Success and failure of moment equations will be highlighted, and they will be used to interpret the sometimes unexpected flow and temperature patterns.

*Ilya Karlin (ETH Zürich (Switzerland))*

#### Abstract

In the first part of this talk, a review of exact and non-perturbative
derivations of hydrodynamics from Boltzmann and similar kinetic systems will be
presented [1].

In the second part, a derivation of the dynamic correction to Grad’s moment
system from kinetic equations (regularized Grad’s thirteen moment system, or R13) is
revisited [2].

The R13 distribution function is found as a superposition of eight modes. Three
primary modes, known from the previous derivation [Karlin et al, Phys. Rev. E57,
**1668** (1998)], are extended into the nonlinear parameter domain. Three
essentially nonlinear modes are identified, and two ghost modes which do not
contribute to the R13 fluxes are revealed. The eight-mode structure of the R13
distribution function implies partition of R13 fluxes into two types of
contributions: dissipative fluxes (both linear and nonlinear) and nonlinear
streamline convective fluxes.

Physical interpretation of the latter non-dissipative and non-local in time effect is discussed,
and some further extensions of R13 are proposed.

#### References

[1] A.N. Gorban and I. Karlin,
*Hilbert’s 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations*,
Bull. Amer. Math. Soc. **51** (2), 187-246 (2014)

[2] I. Karlin,
*Derivation of regularized Grad’s moment system from kinetic equations: Modes, ghosts and
non-Markov fluxes*,
Phil. Trans. R. Soc. A, in press (2018)

*Marshall Slemrod (University of Wisconsin, Madison (U.S.A.) & Weizmann Institute of Science, Rehovot (Israel))*

#### Abstract

In 1900 David Hilbert in his famous ICM address gave a list of problems for the 20th century. His 6th problem raised the issue of a rigorous passage from the “atomistic” theory of materials (say a gas) to a continuum theory. Specifically Hilbert suggested the use of the “atomistic” theory of his day (Boltzmann’s kinetic theory of gases) to pass to the continuum Euler equations of gas dynamics. In this talk I’ll give a short summary of my results suggesting there is an obstruction to this limiting process and that Hilbert’s program will fail in general.

*Markus Hütter (Eindhoven University of Technology (The Netherlands))*

#### Abstract

This presentation discusses three aspects of nonequilibrium-thermodynamic (NET) modeling:

(1) *Jacobi identity*. Some NET frameworks impose severe restrictions on the
reversible dynamics of the system, namely the bracket formalism [1] and the
GENERIC [2-4]. An essential part of these is the so-called Jacobi identity,
representing the time-structure invariance of the reversible dynamics. In this
presentation, different strategies (incl. do’s and don’t’s) for the verification
of the Jacobi identity are discussed. Can we shape and amend this bouquet of
strategies into a workable procedure for ‘non-specialists’?

(2) *Nonlinear force-flux relations*. When it comes to modeling irreversible
dynamics, one is basically faced with the need to specify relations between
thermodynamic forces and thermodynamic fluxes. Different approaches have been
taken to that end, namely one based on a dissipation potential, and another
so-called quasi-linear one. It is shown that the quasi-linear approach is more
general than the one using a dissipation potential [5].

(3) *Mean-field effects*. In order to describe the behavior of a complex system
on a certain scale, a two-scale model can be set-up, in which fine-scale
features are taken into account on the coarser scale. NET is used as a guardrail
in this model formulation. Often, the fine-scale description encompasses a
distribution function of fine-scale degrees of freedom. Typically, the two-scale
coupling involves (ensemble-)averaging of fine-scale systems, where the quantity
to be averaged is defined purely on the basis of each individual fine-scale
system. This enables simulations of the CONNFFESSIT-type [6]. In some cases,
however, there are cross-couplings between the fine-scale systems that make the
use of the CONNFFESSIT-approach impossible. A specific example will be presented
in which such cross-coupling is present, and the numerical difficulties that
arise from this are discussed [7].

#### References

[1] Beris, A. N., Edwards, B. J., 1994.
*Thermodynamics of Flowing Systems*.
Oxford Science Publications.

[2] Grmela, M., Öttinger, H. C., 1997.
Phys. Rev. E, **56**, 6620–6632.

[3] Öttinger, H. C., Grmela, M., 1997.
Phys. Rev. E, **56**, 6633–6655.

[4] Öttinger, H. C., 2005.
*Beyond Equilibrium Thermodynamics*.
Wiley Interscience.

[5] Hütter, M., Svendsen, B., 2013.
Continuum Mech. Thermodyn., **25**, 803–816.

[6] Laso, M., Öttinger, H. C., 1993.
J. Non-Newton. Fluid Mech., **47**, 1–20.

[7] Semkiv, M., Anderson, P. D., Hütter, M., 2017.
J. Comput. Phys., **350**, 184–206.

*Bob Svendsen (RWTH Aachen University & Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf (Germany))*

#### Abstract

The purpose of the current work is the formulation of non-isothermal thermodynamic models for chemically and structurally inhomogeneous solids undergoing finite deformation. In particular, these models are based on both “standard” non-equilibrium thermodynamics [SNET: 1] and on the General Equation for Non-Equilibrium Reversible-Irreversible Coupling [GENERIC: 2]. In particular, model formulations are obtained in this context in [3] for non-isothermal generalizations of standard isothermal conservative (e.g., Cahn-Hilliard) and non-conservative (e.g., Allen-Cahn) diffuse interface or “phase-field” models for multicomponent, multiphase solids. In the context of noflux boundary conditions, the SNET- and GENERIC-based approaches are shown to be completely consistent with each other and result in equivalent temperature evolution relations. The current treatment is consistent with and subsumes previous work on non-isothermal systems [e.g., 4] and GENERIC-based formulation of the Ginzburg-Landau equation [e.g., 5]. As shown by the current work, non-standard and higher-order gradients and fluxes postulated on purely formal grounds [e.g., 6, 7] are completely superfluous. This is due in particular to the treatment of boundary conditions in the current approach as part of the physical model formulation. Indeed, the choice of these has a direct influence on model behavior and predictions (i.e., via solution of the evolution / field relations), especially in the non-local case. For more details, the reader is referred to [3].

#### References

[1] M. Šilhavý,
*The Mechanics and Thermodynamics of Continuous Media*,
Springer Verlag, Berlin, 1997.

[2] H. C. Öttinger,
*Beyond Equilibrium Thermodynamics*,
Wiley Interscience, New York, 2005.

[3] S. Gladkov, J. Kochmann, M. Hütter, S. Reese and B. Svendsen,
*Thermodynamic model formulations for inhomogeneous solids with application to
non-isothermal phase field modeling*,
J. Non-Equilib. Thermodyn. vol. **41**(2), pp. 131–139, 2016.

[4] O. Penrose and P. C. Fife,
*On the relation between the standard phase-field model and a “thermodynamically
consistent” phase-field model*,
Physica D, vol. **69**, pp. 107–113.

[5] A. Mielke,
*Formulation of thermoelastic dissipative material behaviour using GENERIC*,
Cont. Mech. Thermodyn. vol. **23**, pp. 233–256, 2011.

[6] M. Gurtin,
*Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance*,
Physica D vol. **92**, pp. 178–192, 1996.

[7] M. Maraldi, G. N. Wells and L. Molari,
*Phase field model for coupled displacive and diffusive microstructural processes under
thermal loading*,
J. Mech. Phys. Sols. vol. **59**, pp. 1596–1612, 2011.

*Patrick Ilg (University of Reading (United Kingdom))*

#### Abstract

Complex fluids respond sensitively to external perturbations, typically leading to anisotropic macroscopic properties. On a microscopic level, soft matter systems are successfully described by interacting many-particle models obeying Hamiltonian dynamics for translational and rotational degrees of freedom. In order to capture the long-time, large-length scale behavior, coarse-grained descriptions of the orientational ordering and dynamics are urgently needed. Here, we discuss a general method based on projection operators and nonequilibrium statistical thermodynamics in order to systematically and consistently connect microscopic and coarse-grained levels of description. We demonstrate the method for the example of nematic liquid crystals and show its benefits and limitations.

*Athanasios Tzavaras (King Abdullah University of Science and Technology (Saudi Arabia))*

#### Abstract

The relative entropy is a calculation originally developed for hyperbolic systems of conservation laws by Dafermos and DiPerna, which exploits the thermodynamical entropy structure of hyperbolic systems in order to compare two appropriate solutions of the same or related thermomechanical systems. In a recent development a similar calculation is developed for problems associated to a variational structure. In this talk I will survey the method in two directions:

(a) For entropy dissipating hyperbolic-parabolic systems where the hyperbolic part is symmetrizable. This indicates the role of the second law of thermodynamics and as an application provides convergence from the system of thermoviscoelasticity to the system of adiabatic elasticity in the limitas the viscosity and heat conduction tend to zero for smooth solutions.

(b) A class of dispersive systems that can be written as Euler flows generated by a variational structure induced by an energy functional. This class admits as examples the Euler-Korteweg system, the quantum hydrodynamics system, and the Euler-Poisson system. For these problems we develop a relative energy identity which in turn yields various asymptotic convergence results again for smooth solutions.

*Michael Westdickenberg (RWTH Aachen University (Germany))*

#### Abstract

We discuss a variational principle in the spirit of generalized gradient flows (steepest descent) for the compressible Euler equations of gas dynamics.

*Alexander Mielke (WIAS Berlin & HU Berlin (Germany))*

#### Abstract

A gradient structure for an evolutionary system is a triple, a so-called gradient system, consisting of a state space, a possibly time-dependent energy functional, and a dissipation potential that induces the given evolutionary system. The gradient structure, which is not unique, contains additional thermodynamical information, e.g. on the fluctuations in an associated microscopic model.

Considering a family of gradient systems depending on a small parameter, it is natural to ask for the limiting or effective gradient system if the parameter tends to 0. While there is extensive work on the derivation of the effective evolutionary systems, we want to propose some ideas towards the derivation of the effective gradient structure. For the effective energy it is natural to use the Gamma limit, while for deriving the effective dissipation potential we use De Giorgi’s Energy-Dissipation Principle (EDP). We discuss several versions of EDP convergence and show by some examples that the theory is flexible enough to allow situations where starting from quadratic dissipation potentials we arrive at effective dissipation potentials that are no longer quadratic, as in dry friction.

*László Székelyhidi Jr. (University of Leipzig (Germany))*

#### Abstract

In the talk we present the technique of convex integration for constructing weak solutions to various equations in fluid mechanics. We will focus on the recent resolution of Onsager’s conjecture, but also discuss further directions and in particular the applicability to dissipative systems.

*Johannes Zimmer (University of Bath (United Kingdom))*

#### Abstract

Fluctuations can often provide useful information on underlying processes, for example in form of fluctuation-dissipation relations. We present some results, old and new, and as well as applications, such as the determination of transport coefficients. This is joint work with P. Embacher, N. Dirr, C. Reina, R. Jack and M. Kaiser.

*Matteo Polettini (Université du Luxembourg)*

#### Abstract

Ever since Einstein’s description of Brownian motion, the mathematics of Markovian stochastic processes proved a crucial paradigm for the understanding of the behaviour of open systems interacting with a large (and thus memory-less) environment. When such processes occur on an intrinsically discrete configuration space, the topology of the network of possible transitions furnishes important insights about the nature of nonequilibrium processes. In this talk I will 1) trace the conceptual path that leads from the abstraction of open systems to the mathematics of Markov processes on networks, 2) mention some of the major findings that are novel with respect to textbook thermodynamics, 3) outline some recent developments regarding the role of an ideal observer who only has partial information about the system.

*Alberto Montefusco (ETH Zürich (Switzerland))*

#### Abstract

“The mathematics of statistical mechanics is large-deviation theory” (Hugo Touchette). While this statement is true for equilibrium statistical mechanics, its application to dynamics has not been fully achieved yet. In this talk, I illustrate a one-to-one connection between a class of reversible Markov processes and a class of deterministic systems (the “generalised gradient flows”), which I interpret as a generalisation of the Onsager-Machlup theory of fluctuations or of the fluctuation-dissipation relation of the second kind. I exploit this connection to develop a new method of coarse-graining and test it in the case of a reduction from the diffusion in a double-well potential to the jump process that describes a monomolecular reaction (the Kramers’ escape problem).

*Gian Paolo Beretta (Università di Brescia (Italy))*

#### Abstract

We have shown in [1] that the dissipative (irreversible) structure of several mathematical frameworks for modeling nonequilibrium states and their dynamics can be unified by extending the idea of steepest entropy ascent (SEA) that we first introduced in [2] in quantum thermodynamics (QT) modeling. Such frameworks include: Small-Scale and Rarefied Gases Dynamics (Boltzmann equation and its kinetic models); Rational Extended Thermodynamics, Macroscopic Non-Equilibrium Thermodynamics, and Chemical Kinetics; Mesoscopic Non-Equilibrium Thermodynamics; Quantum Computing [3]. In [4] we proved that the dissipative part of the nonequilibrium formulation known as GENERIC is essentially an implementation of the SEA principle: whenever the GENERIC structure is equipped with an inner product, the SEA and GENERIC models of the irreversible component of the dynamics are essentially interchangeable, provided of course they assume the same kinematics. The talk will show as case study the use of SEA to develop models of the Boltzmann equation [5].

In [6] SEAQT has been used to model effectively quantum decoherence. By combining the SEA principle in the QT framework with a constrained-equilibrium (or hypoequilibrium) model reduction approach, Ref. [7] developed the SEAQT approach into a quantum statistical thermodynamic analysis of nonequilibrium evolution, especially effective in providing far-from-equilibrium time-dependent information also for macroscopic and mesoscopic systems. Rather than from a phenomenological description, the method starts from a more fundamental density-of-states representation of existing models of the system’s eigenstructure (see e.g. [8]).

#### References (in the pdf version click on the title to get to the paper online)

[1] G.P. Beretta,
[*Steepest entropy ascent model for far-nonequilibrium thermodynamics: Unified implementation of the maximum entropy production principle*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/m44-GPBeretta-PhysRevE-90-042113-2014.pdf),
Phys. Rev. E, Vol. **90**, 042113 (2014).

[2] G.P. Beretta,
[*Steepest entropy ascent in quantum thermodynamics*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/m10-Beretta-LectureNotesPhys-278-441-1987.pdf),
in *The Physics of Phase Space*, Lecture Notes in Physics, (Springer-Verlag, Berlin),
edited by Y.S. Kim and W.W. Zachary, Vol. 278, 441-443 (1986).**See also:**

G.P. Beretta, E.P. Gyftopoulos, and J.L. Park,
[*Quantum thermodynamics. A new equation of motion for a general quantum system*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/m06-BerettaGyftopoulosPark-NuovoCimento-87-77-1985.pdf),
Nuovo Cimento B, Vol. **87**, 77-97 (1985).

John Maddox,
Nature,
Vol. 316, 11 (1985).

G.P. Beretta,
[*Quantum thermodynamics of nonequilibrium. Onsager reciprocity and dispersion-dissipation relations*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/m11-Beretta-FoundPhys-17-365-1987.pdf),
Found. Phys., Vol. **17**, 365-381 (1987).

G.P. Beretta,
[*Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/m34-Beretta-ROMP-64-139-2009.pdf),
Reps. Math. Phys., Vol. **64**, 139-168 (2009). http://dx.doi.org/10.1016/S0034-4877(09)90024-6

[3] F. Tabakin,
[*Model dynamics for quantum computing*]
(http://quantum-thermodynamics.unibs.it/Tabakin-AnnPhys-383-33-2017.pdf),
Ann. Phys., Vol. **383**, 33 (2017).

[4] A. Montefusco, F. Consonni, and G.P. Beretta,
[*Essential equivalence of the GENERIC and Steepest Entropy Ascent models of dissipation for non-equilibrium thermodynamics*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/m54-MontefuscoConsonniBeretta-PhysRevE-91-042138-2015.pdf),
Phys. Rev. E, Vol. **91**, 042138 (2015).

[5] G.P. Beretta and N.G. Hadjiconstantinou,
[*Steepest entropy ascent models of the Boltzmann equation. Comparisons with hard-sphere dynamics and relaxation-time models for homogeneous relaxation from highly non-equilibrium states*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/ic67-BerettaHadjiconstantinou-ASME-IMECE2013-64905.pdf),
Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition,
paper IMECE2013-64905, November 15-21, 2013, San Diego, CA, Proc. ASME 56352;
Vol. 8B: *Heat Transfer and Thermal Engineering*, V08BT09A050,
http://dx.doi.org/10.1115/IMECE2013-64905

[6] S. Cano-Andrade, G.P. Beretta, and M.R. von Spakovsky,
[*Steepest-entropy-ascent quantum thermodynamic modeling of decoherence in two different microscopic composite systems*]
(http://gianpaolo-beretta.unibs.it/Beretta-papers-online/m53-CanoAndradeBerettaVonSpakovsky-PhysRevA-91-013848-2015.pdf),
Phys. Rev. A, Vol. **91**, 013848 (2015).

[7] G. Li and M.R. von Spakovsky,
[*Steepest-entropy-ascent quantum thermodynamic modeling of the relaxation process of isolated chemically reactive systems using density of states and the concept of hypoequilibrium state*]
(http://quantum-thermodynamics.unibs.it/LiVonSpakovsky-PRE-93-012137-withErrata.pdf),
Phys. Rev. E, Vol. **93**, 012137 (2016).

[8] G. Li, M.R. von Spakovsky, and C. Hin,
[*Steepest entropy ascent quantum thermodynamic model of electron and phonon transport*]
(http://quantum-thermodynamics.unibs.it/LiVonSpakovskyHin-PhysRevB-97-024308-2018.pdf),
Phys. Rev. B, Vol. **97**, 024308 (2018).

*Hans-Christian Öttinger (ETH Zürich (Switzerland))*

#### Abstract

The most important tasks of nonequilibrium thermodynamics are formulated in the form of ten commandments. Several of these tasks are deeply related to mathematics. The ten commandments are presented to stimulate critical discussions and selected issues are elaborated in some detail, where the focus is on the structure of thermodynamically admissible equations and on statistical mechanics.

The objective of the workshop will be to bring together experts from mathematics and the natural sciences, to discuss recent developments in non-equilibrium thermodynamics and related topics. The aim is to open up a dialogue and explore possible connections, bringing together the respective points of view of the different disciplines.

The workshop will consist of short presentations of 40 minutes, followed by ample time for discussions. The presentations are not intended to address the most recent results, but instead will provide a condensed summary (the “big picture”) of the main ideas and techniques, possibly illustrated by a representative example. Further details can be explored in the discussions. We hope that such a format will offer an overview over recent trends and new developments, and a possibility to compare different approaches and techniques.

- Time
- March 5-7, 2018 (Mon-Wed)
- Location
- RWTH Aachen University, Lecture Hall V

To register for the event, please send an email to info@eddy.rwth-aachen.de.

There is no conference fee.

## Program

The workshop will take place in Lecture Hall V in the main building of RWTH Aachen University (Templergraben 55, 52062 Aachen).

### Monday, March 5

10:00-10:40h

Hans-Christian Öttinger

Polymer Physics, ETH Zürich (Switzerland)**The Ten Commandments of Nonequilibrium Thermodynamics**

Abstract

11:00-11:40h

Gian Paolo Beretta

Fluid and Thermal Sciences, University of Brescia (Italy)**Steepest Entropy Ascent model for far-non-equilibrium dissipative evolution in quantum thermodynamics: a 1984 precursor of GENERIC (1997), gradient flows (1998), maximal entropy production (2001), and SEAQT (2014)**

Abstract

Lunch Break

14:00-14:40h

Alberto Montefusco

Polymer Physics, ETH Zürich (Switzerland)**Coarse-graining via large-deviation theory**

Abstract

15:00-15:40h

Matteo Polettini

Physics and Materials, Université du Luxembourg**Network methods for nonequilibrium thermodynamics**

Abstract

16:00-16:40h

Johannes Zimmer

Mathematical Science, University of Bath (United Kingdom)**Fluctuations in processes out of equilibrium**

Abstract

### Tuesday, March 6

9:00-9:40h

László Székelyhidi Jr.

Mathematics, University of Leipzig (Germany)**Convex integration in fluid dynamics**

Abstract

10:00-10:40h

Alexander Mielke

Mathematics, WIAS Berlin & HU Berlin (Germany)**Construction of limiting gradient systems via EDP convergence**

Abstract

11:00-11:40h

Michael Westdickenberg

Mathematics, RWTH Aachen University (Germany)**A variational principle for compressible Euler equations**

Abstract

Lunch Break

14:00-14:40h

Athanasios Tzavaras

Mathematics, King Abdullah University of Science and Technology (Saudi Arabia)**Relative entropy and applications to the convergence of thermomechanical theories**

Abstract

15:00-15:40h

Patrick Ilg

Theory of Soft Matter, University of Reading (United Kingdom)**Macroscopic dynamics of complex fluids from systematic coarse graining their molecular dynamics**

Abstract

16:00-16:40h

Bob Svendsen

Material Mechanics, RWTH Aachen University & Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf (Germany)**Formulation of non-local, non-isothermal models for inhomogeneous solids**

Abstract

17:00-17:40h

Markus Hütter

Mechanical Engineering, TU Eindhoven (The Netherlands)**Comments about two theoretical issues, and a numerical challenge**

Abstract

### Wednesday, March 7

9:00-9:40h

Marshall Slemrod

Mathematics, University of Wisconsin-Madison (U.S.A.) & Weizmann Institute of Science, Rehovot (Israel)**Hilbert’s 6th Problem and the Failure of the Boltzmann to Euler Limit**

Abstract

10:00-10:40h

Ilya Karlin

Computational Kinetics, ETH Zürich (Switzerland)**Hydrodynamic manifolds from kinetic equations**

Abstract

11:00-11:40h

Henning Struchtrup

Mechanical Engineering, University of Victoria (Canada)**The Moment Method in Kinetic Theory: Successes and Challenges**

Abstract

12:00-12:40h

Manuel Torrilhon

Mathematics, RWTH Aachen University (Germany)**Hierarchical Boltzmann simulations and m(odel)-refinement**

Abstract

## Travel

### Accommodation

Hotel rooms in the center of Aachen can be booked through the online registration form available at the following link
(which takes you to the **aachen tourist service e.v.** website):

Upon booking you will receive a confirmation email. The rates listed there are per room and per day, and include taxes. Breakfast may or may not be included. There are no additional registration fees. The booked rooms will be available from the first night of the reservation.

### How to get to Aachen…

Aachen is well connected to the train system in Germany. Tickets can be purchased online at Deutsche Bahn. There are fast train connections to Paris, Brussels, and London operated by Thalys in combination with Eurostar.

- From Frankfurt Airport
- The train ride from Frankfurt to Aachen takes about two to three hours, depending on the connections, with a changeover at Cologne Main Train Station. It costs between 51 and 74 Euros, depending on if you take a regional train with stops in between or quicker ICE connection with the Deutsche Bahn. Tickets can be acquired in the train office at the airport with either cash or credit card.
- From Düsseldorf Airport
- Depending on the connections, the train ride from Düsseldorf to Aachen takes about one and a half to two hours with the Deutsche Bahn. Some connections require a changeover at Düseldorf Main Train Station or Cologne Main Train Station. Costs range from 18-28 Euros. Tickets can be bought in the train office at the airport with either cash or credit card.
- Fom Cologne/Bonn Airport
- Depending on the connections, the train ride with Deutsche Bahn
from Cologne/Bonn Airport to Aachen takes about one to one and half hours with a changeover at Cologne Main Train Station.
Costs range from 15-21 Euros.
Tickets can be bought in the train office at the airport with either cash or credit card.

## Organizers

Mark A. Peletier

Mathematics, TU Eindhoven (The Netherlands)

Manuel Torrilhon

Mathematics, RWTH Aachen University (Germany)

Athanasios Tzavaras

Mathematics, King Abdullah University of Science and Technology (Saudi Arabia)

Michael Westdickenberg

Mathematics, RWTH Aachen University (Germany)

## Sponsors

RWTH Aachen University

Deutsche Forschungsgemeinschaft (DFG)

EU-ITN ModCompShock

*Tabea Tscherpel (University of Oxford (U.K.))*

#### Abstract

The object of this talk is a class of generalised Newtonian fluids with implicit
constitutive law. Both in the steady and the unsteady case, existence of weak
solutions was proven by Bulíček et al. (2009, 2012), and the main challenge is
the small growth exponent $q$ and the implicit law. I will discuss the
application of a splitting and regularising strategy to show convergence of
finite element approximations to a weak solution of the flow. In both the steady
and the unsteady case this allows us to cover the full range of growth
exponents. Thus, it generalises existing work of Diening et al. (2013) in the
steady case, and in the unsteady case it provides a new result in the framework
of implicitly constituted fluids.

This is joint work with Endre Süli.

*Gabriella Puppo (Università degli studi dell'Insubria (Italy))*

#### Abstract

Starting from a short summary on the structure of kinetic theory for a single
gas, I’ll describe a technique for constructing models for mixtures of gases. In
particular, I’ll construct a model based on the BGK approximation, computing the
exchange terms, and proving an H-theorem. The numerical treatment of these
models will be considered, introducing an Asymptotic Preserving scheme, which
ensures that the correct asymptotic regime is recovered also by the numerical
solution.

I will end with an application of the same ideas to traffic flow.