Workshop on PDEs: Modelling,
Analysis and Numerical Simulation

# PDE-MANS 2020

## Program for the Workshop

 Monday 13 Tuesday 14 Wednesday 15 Thursday 16 9:30 - 10:15 Merino Tang Bruna Evans 10:15 - 11:00 Gabriel Chalons Fagioli Gvalani 11:00 - 11:30 Coffee Coffee-Poster Session Coffee-Poster Session Coffee 11:30 - 12:15 Calsina Semplice Esposito Yoldaş 12:15 - 13:00 Toscani Vecil Menegaki Guerand Poster Session Poster Session Lods 15:00 - 15:45 De Pittà Figueroa Desvillettes 15:45 - 16:30 Pouradier-Duteil Solem Jankowiak

## Talks

Excluded-volume and order in systems of Brownian needles
Maria Bruna

In this talk we study a system of (hard-core) interacting Brownian needles. Unlike point particles, the finite size and shape of each needles has an influence on the evolution of the system. We explore the effects of excluded volume and anisotropy at the population level for these systems. Since needles exclude less volume if aligned, can excluded-volume effects alone induce order in the system? Starting from the stochastic particle system, we derive an integro-differential equation for the population density using the method of matched asymptotic expansions and conformal mappings. We present numerical simulations of the particle- and population-level models, and discuss the limit of large rotational diffusion.

On R0 for populations with continuous structure
Àngel Calsina

In a model of population growth, the basic reproduction number R0 is defined as the expected number of children that an individual has throughout his life in a fixed environment (in an epidemiological model, as the expected number of new infections a newly infected individual will produce).

In structured populations, when the birth event can happen in different individual states (of size, phenotype, spatial position, etc.) one talks of “typical” individual but it is not always clear what “typical” individual means and so, which is the expected number of children of a ”typical” individual.

On the other hand, in deterministic modeling we could be unsatisfied with a definition that involves a probability concept as expectation. Both drawbacks are solved defining R0 as the spectral radius of the so-called first- generation operator, which maps a (distribution of) population to the (distribution of) population of their children along the whole life span of the former ([3]).

Nevertheless, also this definition has some drawback since in some models the definition of the birth operator and that of the first-generator operator depend on the choice of the sometimes arbitrary concept of birth event ([2], [1]).

When the structuring variable is discrete, i.e. when the population dynamics can be described by a system of odes, the first-generation operator is a matrix and there is no problem with the definition above leaving apart that it may be non-uniquely determined. However, in continuously structured populations with concentrated state at birth, the birth rate shows as a boundary condition instead of a bounded (birth) operator. This makes difficult to adapt to Banach spaces the above definition. We consider some examples where R0 is defined as a limit of basic reproduction numbers of approximate models with distributed states at births for which the next generation operator can be defined as a bounded linear operator.

1. Barril C., Calsina À., Ripoll J. On the reproduction number of a gut microbiota model. Bull. Math. Biol. 79 (2017)
2. Cushing, J. M, Diekmann, O. The many guises of R0 (a didactic note). J. Theoret. Biol. 404 (2016)
3. O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts. The construction of next generation matrices for compartmental epidemic models. J. R. Soc. Interface 7 (2010)

High-order fully well-balanced Lagrange-Projection scheme for shallow water equations
Cristophe Chalons

We will discuss the derivation of high-order and well-balanced Lagrange-Projection schemes for shallow-water equations. As we will see, the Lagrange-Projection strategy is especially well adapted in low Mach or low Froud regimes.

Mean-field analysis of history-dependent, stochastic activity in neuron-glial networks
Maurizio De Pittà

The activity of neurons in the brain randomly fluctuates as a result of mechanisms of noise that can be born in the stimuli impinging on neurons from the external environment, or that are inherent of the cellular nature. A typical description of neural networks of the brain does not consider glial signaling which however has the potential to regulate connections between neurons in an activity-dependent fashion. This scenario considerably complicates the solution of the first passage time problem of neuronal firing activity, often making the underpinning Fokker-Planck approach analytically intractable. We present mathematical arguments to overcome this apparent conundrum, providing insights on the phase portrait of a mean-field description of history-dependent activity of neuron-glial networks. The approach is general and can be extended to other examples of complex heterogeneous networks in the context of biological systems and beyond.

A chemotaxis model treated with duality methods
Laurent Desvillettes

In a work in collaboration with Yong-Jung Kim, Ariane Trescases and Changwook Yoon, we introduce a chemotaxis model featuring at the same time aggregation phenomena and global existence. This last property of the model is obtained (in dimension 1) thanks to the use of duality lemmas, such as those developed for the treatment of cross diffusion problems coming out of population dynamics.

Nonlocal-interaction equation on graphs
Antonio Esposito

I will discuss a recent result obtained in collaboration with F. S. Patacchini, A. Schlichting, and D. Slepcev. We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou--Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of vertices'' is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$^2$IE). We develop the existence theory for the solutions of the NL$^2$IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$^2$IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

Quantitative Rates of Convergence to Equilibrium for equations with spatially degenerate jump rates
Josephine Evans

I will discuss kinetic equations whose jump rate $\sigma(x)$ depends on the position in space. Exponential convergence to equilibrium for such equations was proved to be equivalent to a geometric control condition by Bernard and Salvarani for the zero potential case and then by Han-Kwan and Leautaud for more complex jump kernels and with non zero potential. These works do not give quantitative rates. I will explain how Doeblin's theorem can be used to give quantitative rates of convergence to equilibrium.

Nonlinear diffusion equations with degenerate mobilities
Simone Fagioli

I will present some results on aggregation/diffusion equations arising in some applied context, such as modeling opinion formation phenomena. Those equations describe the dynamics for a certain density population and are composed of nonlinear diffusion equations with degenerate mobility combined with other effects, such as transport driven by external forces (local potentials) and/or aggregation or repulsion induced by the presence of non-local potentials. The equations are posed on a bounded real interval. In case of fast-decay mobilities, namely mobilities functions under Osgood integrability condition, a suitable coordinate transformation is introduced. We observe that the coordinate transformation induces a mass-preserving scaling on the density and we show that the rescaled density is the unique weak solution to a nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the new density allow us to motivate the aforementioned change of variable and to state the results in terms of the original density without prescribing any boundary conditions. This is a joint work with N. Ansini. In the case of slow decay mobility we use a different approach, namely solutions are obtained as large particle limit of a suitable nonlocal version of the Follow-the-Leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. This approach is also used to show existence in the multiple species opinion formation case (e.g. opinion leaders and followers) and to study the long time behavior both analytically and numerically. This is a joint work with E. Radici.

Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches
Susely Figueroa Iglesias

Abstract: Horizontal Gene Transfer (HGT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HGT is believed to be the main instrument of developing the antibiotic resistance. In this talk, I consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integro-differential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal will be to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular.

Irreducibility, aperiodicity, and coupling for a nonlocal population dynamics model
Pierre Gabriel

We consider a linear nonlocal equation with drift which appears in selection-mutation and reaction-diffusion models in a shifting environment. We obtain the geometric ergodicity of this equation by introducing a new irreducibility-aperiodicity type condition inspired from the coupling method in probabilities. The talk is based on joint works with Vincent Bansaye, Bertrand Cloez, and Aline Marguet.

Quantitative regularity for parabolic De Giorgi classes
Jessica Guerand

De Giorgi method is a way to prove Hölder regularity of solutions of parabolic equations. While in the elliptic case the proof is completely quantitative, in the parabolic case it seems to remain a non-quantitative step: the intermediate value lemma. The purpose of this talk is to present a quantitative version of this step after introducing how it is useful to get Hölder regularity.

Periodic homogenisation and the mean field limit for weakly interacting diffusions
Rishabh Gvalani

We analyse the statistical behavior of a large number of weakly interacting diffusion processes with highly oscillatory periodic interaction potentials. We study the combined limit of taking the number of particles to infinity, also known as the mean field limit, and taking the period of the potential to zero, also known as the homogenisation limit. In particular, we show that these limits do not commute if the quotiented process undergoes a phase transition, that is to say if it admits more than one invariant measure. As an incidental, we analyze the energetic consequences of the classical fluctuation central limit theorem and derive optimal rates of convergence of the Gibbs measure to the unique limit of the mean field energy in relative entropy. Joint work with Matias Delgadino and Greg Pavliotis.

Cell motility without adhesion: mathematical and numerical modelling
Gaspard Jankowiak

A number of cells, including leukocytes, crawl on surfaces by adhesion, creating local bonds between the substrate and the cytoskeleton. In recent experiments, biologists studied how such cells travel in artificial microchannels, before and after engineering them to prevent adhesion. If the channel walls are smooth, non adhering cells stand still. On the contrary however, if they present structures of adequate size, motility is recovered, with speeds comparable to adhering cells. This raises the question of mechanisms other than adhesion, which can allow movement.

I will discuss the experiments briefly, and introduce a simplified mechanical model describing this behavior, based on elementary physical considerations. Two key ingredients allow for forward motion: the first is the renewal of the actin cortex through polymerization, which results in mass being transported to the front. This require internal forces to act, the reaction of which also weighs on the surrounding channel wall, thanks to the second ingredient: the cortex’s internal rigidity. The resulting system of parabolic PDEs, which can be analyzed partially, and existence results will be given in simple situations.

I will also discuss some numerical experiments and an extension of the model which includes the mechanical contribution of the cell’s nucleus.

This is a joint work with C. Schmeiser, D. Peurichard, L. Preziosi and C. Giverso

Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials
Bertrand Lods

In this joint wok with Ricardo Alonso (PUC-Rio, Brazil) and Véronique Bagland (Université Clermont-Auvergence, France) we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid for rather general initial datum. An important feature of the estimates is the independence with respect to the quantum parameter. Consequently, in the classical limit the same estimates are recovered for the Landau equation.

Quantitative Rates of Convergence to Non-Equilibrium Steady States for the Chain of Oscillators
Angeliki Menegaki

A long-standing issue in the study of out-of-equilibrium systems in statistical mechanics is the validity of Fourier's law. In this talk we will present a model introduced for this purpose, i.e. to describe properly heat diffusion. It consists of a $1$-dimensional chain of $N$ oscillators coupled at its ends to heat baths at different temperatures.
Here, working with a weakly anharmonic homogeneous chain, we will show how it is possible to prove exponential convergence to the non-equilibrium steady state in Wasserstein-2 distance and in Relative Entropy. The method we follow is a generalised version of the theory of $\Gamma$ calculus. It has the advantage to give quantitative results, thus we will discuss how the convergence rates depend on the number of the particles $N$. If time permits, we will also talk about the optimal spectral gap for harmonic (homogeneous or not) oscillators chains, using different techniques that was obtained in a joint work with Simon Becker.

Nematic alignment of self-propelled particles in the macroscopic regime.
Sara Merino

We consider a model for collective dynamics where particles move at a constant speed and change their direction of motion to align with the direction of motion of their neighbours, up to some noise. In particular, particles may align in the same or opposite orientation; this is called nematic alignment. Starting from this particle model we derive a macroscopic model for the particle density and mean direction of motion which corresponds to a cross-diffusion system. The derivation is carried over by means of a Hilbert expansion. This cross diffusion system poses many new challenging questions.
This is a joint work with Pierre Degond (Imperial College London, UK).

Sparse control of Hegselmann-Krause models: Black hole and declustering.

We elaborate control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions that characterize whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided).

Central WENO reconstructions: from their origin to the most recent developments and applications.
Matteo Semplice

In this seminar I will review development of Central WENO reconstructions. They were first introduced to construct a third order staggered scheme, but, much later, many advantages have been noticed over standard WENO procedures in numerical schemes for conservation and balance laws, in one and more space dimensions, uniform and non-uniform grids. I will focus on their most recent formulations for finite volume schems, that include WENOZ-type nonlinear weights, for which a thorough analysis of the global smoothness indicator $\tau$ is now available.

In this second part of the talk I will present several examples of numerical schemes where, due to the non-uniform/non-conforming grid or the special discretizations required for well-balancing, CWENOZ reconstructions best show their benefits.

1. I. Cravero, G. Puppo, M. Semplice, G. Visconti. CWENO: uniformly accurate reconstructions for balance laws. Math. Comp. (2018). DOI 10.1090/mcom/3273
2. I. Cravero, G. Puppo, M. Semplice, G. Visconti. Cool WENO schemes Comp. & Fluids 169:71--86 (2018). DOI 10.1016/j.compfluid.2017.07.022
3. I. Cravero, M. Semplice, G. Visconti. Optimal definition of the nonlinear weights in multidimensional Central WENOZ reconstructions. SINUM (2019). DOI 10.1137/18M1228232
4. M. Semplice, A. Coco, G. Russo. Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction. J. Sci. Comput. (2016). DOI 10.1007/s10915-015-0038-z
5. C. Klingenberg, G. Puppo, M. Semplice. Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity. SIAM J. Sci. Comput. (2019). DOI 10.1137/18M1196704
6. M. Castro, M. Semplice. Third- and fourth-order well-balanced schemes for the shallow water equations based on the CWENO reconstruction. Int J Numer Meth Fluids (2019). DOI 10.1002/fld.4700

Complex dynamics in cardiac cells
Susanne Solem

In cardiac muscle cells, arrhythmias which can lead to sudden cardiac death, can occur. It is therefore highly important to understand the exact mechanisms leading to such phenomena. In this talk, the dynamics of two specific Hodgkin—Huxley-type cardiac muscle cell models will be discussed. We will see that these models exhibit complex features related to cardiac arrhythmias, such as chaotic dynamics and early afterdepolarisations (small additional oscillations during an action potential). As the heart consists of a large number of interacting cells, the dynamics of related up-scaled reaction-diffusion-type models will also be presented in order to understand the effects of the discovered features on the macro-scale.

Regularity for reaction-diffusion systems with mass dissipation
Bao Quoc Tang

We present some recent advances on global existence and regularity of reaction-diffusion systems with mass dissipation. These systems typically arise from chemical reactions where spatial diffusion is also present. Under the mass action kinetics, the nonlinearities are usually of polynomial type with arbitrary high orders. This fact makes the global existence of solutions challenging, while the local existence, on the other hand, is classical. It's worth also to mention that the difficulties seem to increase as the spatial dimension n is bigger.

In this talk, we show global existence and uniform-in-time bounds of strong solutions for reaction-diffusion systems with mass dissipation in the following cases: The nonlinearities are at most (slightly super-) quadratic, for all n ≥ 1; The diffusion coefficients are either close to each other, or large enough, for all n ≥ 1; The nonlinearities satisfy a quadratic intermediate sum condition, for n ≤ 2.

This talk is based on joint works with Brian Cupps (Washington), Klemens Fellner (Graz) and Jeff Morgan (Houston).

Statistical description of human addiction phenomena
Giuseppe Toscani

We study the evolution in time of the statistical distribution of some addiction phenomena in a system of individuals. The kinetic approach leads to build up a novel class of Fokker--Planck equations describing relaxation of the probability density solution towards a generalized Gamma density. A qualitative analysis reveals that the relaxation process is very stable, and does not depend on the parameters that measure the main microscopic features of the addiction phenomenon.

Deterministic simulation of a nanoscaled DG-MOSFET on a GPU platform
Francesco Vecil

The DG-MOSFET is a common kind of transistor, the building block of any electronic device. There are advantages in downscaling its size, for both performance and energy-saving goals. The modeling of electrons inside the device is split following the dimension: they are either particles or waves depending on confinement properties. Seven electron-phonon scattering phenomena are taken into account. This gives a physically accurate, high-dimensional model, hence computationally heavy. A fully sequential code might take a few weeks to reach the stationary state. In order to cope with this, in these last years we have first implemented a parallelization on CPU via MPI, then a hybrid parallel model on GPU/CPU via Cuda/OpenMP. We have now reached a version of the code fully implemented on GPU, which has the advantage of avoiding costly exchange of data between the physically distant RAM and DRAM.

Spectral gap for the growth-fragmentation equation via Harris's Theorem
Havva Yoldaş

In this talk, we give a brief explanation of Harris's Theorem and its precursor Doeblin's Theorem which is developed for the study of discrete-time Markov chains. This probabilistic approach is based on quantitatively verifying a minorisation condition and a drift condition in order to obtain quantitative estimates on the long-time behaviour of time evolution for some linear and non-local integro-PDEs. We will see how this method applies to achieve spectral gap for the growth-fragmentation equation which is a linear equation describing a system of growing and dividing particles and may be used as a model for many processes in different contexts like ecology, neuroscience, telecommunications and cell biology.