In this talk we review some classical and recent results
relating the uncertainty principles for the Laplacian with the
controllability and stabilisation of some linear PDEs. The uncertainty
principles for the Fourier transforms state that a square integrable
function cannot be both localised in frequency and space without being
zero, and this can be further quantified resulting in unique
continuation inequalities in the phase spaces. Applying these ideas to
the spectrum of the Laplacian on a compact Riemannian manifold, Lebeau
and Robbiano obtained their celebrated result on the exact
controllability of the heat equation in arbitrarily small time. The
relevant quantitative uncertainty principles known as spectral
inequalities in the literature can be adapted to a number of different
operators, including the Laplace-Beltami operator associated to C^1
metrics or some Schödinger operators with long-range potentials, as we
have shown in recent results in collaboration with Gilles Lebeau (Nice)
and Nicolas Burq (Orsay), with a significant relaxation on the
localisation in space. As a consequence, we obtain a number of
corollaries on the decay rate of damped waves with rough dampings, the
simultaneous controllability of heat equations with different boundary
conditions and the controllability of the heat equation with rough
controls.

In the study of mean-field limits, it is well known that the viscous porous media
equation can be derived from a system of N moderately interacting particles in the limit N → ∞
(Oelschlager ’85 and ’87). However, there are only very few results available concerning the
rate of convergence in the mean-field limit for moderately interacting particles - especially
when we are interested in algebraic rates of convergence which are important in order to show
a central limit theorem for the particles in the moderate setting.
In this talk I will give an introduction to moderately interacting particles and explain how
we can combine the convergence result in L_2 norm by Oelschlager ’87 with relative entropy
methods for mean-field limits developed by Jabin and Wang in order to show a quantitative
propagation of chaos result for the viscous porous medium equation in strong L_1 norm and
algebraic rate N^γ for some γ > 0. Additionally, I will explain the connection between the L_2
-norm convergence result of Oelschlager and fluctuations around the mean-field limit as
well as difficulties that arise for proving Oelschlager’s L_2
-norm result for a more general class
for moderate models.
This is joint work with Li Chen (University of Mannheim) and Xiaokai Huo (Iowa State
University)

We propose a two-species version of the viscoelastic tissue growth model introduced by Perthame and Vauchelet in 2015. We consider two singular limits that establish a link between modelling two paradigms each. The first limit connects the viscoelastic model with a free-boundary descriptions of tissue growth. The second limit is a rigorous justification to pass from Brinkman‘s law to Darcy‘s law (inviscid limit). As a by-product we give a novel existence proof for a degenerate cross-diffusion system with phase-separation property.
The results are based on a sequence of recent works with David, D\k{e}biec, Mandal, Perthame, and Vauchelet.

Abstract: We consider general linear kinetic equations combining
transport with external potential force, linear collision
operator on the kinetic variable, boundary conditions, and
allowing thermalisation degeneracy on part of the spatial
domain. The linear collision operators considered include the
linear Boltzmann and Fokker-Planck operators and the boundary
conditions include specular, diffusive and Maxwell
conditions. We prove quantitative estimates of relaxation to
equilibrium (spectral gap) under a \emph{transport control
condition} generalising previous geometric control
conditions. The argument is new and rely entirely on
trajectories and weighted functional inequalities on the
divergence operators, that are of independent interest and
imply quantitatively weighted Stokes and Korn inequalities.

Kinetic equations play a leading role in the modelling of large systems of interacting
particles/agents with a recognized effectiveness in describing real world phenomena ranging
from plasma physics to multi-agent dynamics. The derivation of these models has often to
deal with physical, or even social, forces that are deduced empirically and of which we have
limited information. To produce realistic descriptions of the underlying systems, it is of
paramount importance to manage efficiently the propagation of uncertain quantities across
the scales.
We concentrate on the interplay of this class of models with collective phenomena in
life and social sciences, where the assessment of uncertainties in data assimilation is crucial
to design efficient interventions. Furthermore, to discuss the mathematical interface of
this class of models with available data, we derive the evolution of observable quantities
based on suitable macroscopic limits of classical kinetic theory. Finally, we analyze
how the introduction of robust control strategies leads to the damping of the uncertainties
characterizing the system at the macroscopic level.

In this talk, I will provide an introduction to generalized gradient structures and variational forms for nonlocal evolutions, and provide various examples of such objects. I will then proceed to cover different cases for which these structures can be conveniently used to prove asymptotic limits via evolutionary Gamma-convergence methods.

We study the validity of the dissipative Aw-Rascle system as a macroscopic model for pedestrian dynamics. The model uses a congestion term (a singular diffusion term) to enforce capacity constraints in the crowd density while inducing a steering behaviour. Furthermore, we introduce a semi-implicit, structure-preserving, and asymptotic-preserving numerical scheme which can handle the numerical solution of the model efficiently. We perform the first numerical simulations of the dissipative Aw-Rascle system in one and two dimensions. We demonstrate the efficiency of the scheme in solving an array of numerical experiments, and we validate the model, ultimately showing that it correctly captures the fundamental diagram of pedestrian flow.

In this talk we shall discuss problems arising in the mathematical description, understanding and advancement of machine learning algorithms. These algorithms find applications in various scientific and engineering domains, significantly impacting key aspects of research. Mathematical analysis is essential to address several crucial questions: a) the reliability of these algorithms, b) their advantages or potential limitations compared to conventional approaches, and c) the design novel and enhanced algorithms. Particular emphasis will be given to the connection of ML algorithms to notions and problems related to PDEs and to Numerical Analysis.

The Lp convergence of eigenfunction expansions on an arbitrary domain is not well understood outside a handful of concrete examples. One way of tackling this problem is to start with a domain, such as the square, where Lp convergence of the expansions is known, and perturb it in a suitable way. We then analyse what happens to the partial sums (or spectral projections) of the eigenfunction expansions after the perturbation.

Continuity results for these spectral projections for the Laplacian on a bounded domain were known for the H_0^1 norm. In 2 dimensions, we prove that a similar continuity result holds in Lp, provided that certain bounds can be obtained on the resolvents of \Delta^{-1}. We also show that these bounds do in fact hold if we begin by perturbing a square or rectangular domain.

This is joint work with James Robinson.

We investigate certain questions arising in two-dimensional statistical hydrodynamics, by relying on principles of entropy maximization for the vorticity of a two-dimensional perfect fluid in a disc. In analogy with the entropy functions used in statistical mechanics and thermodynamics, we show that similar concavity properties hold for the 2d Euler equations when maximizing entropies at fixed energy levels. The proofs rely on rearrangement inequalities, a modification of the classical min-max principle, and the properties of the Euler-Lagrange equations for the corresponding constrained optimization.

In this talk, we present some reflections and recent developments on solving several longstanding open problems in nonlinear conservation laws and related nonlinear partial differential equations through entropy analysis and related methods. Further related topics, perspectives, and open problems will also be addressed.

The objective of the talk is to provide examples of sustained oscillations for hyperbolic-parabolic systems.
This problem was motivated by work on the existence theory for viscoelasticity of Kelvin-Voigt type with non-convex stored energies
(joint with K. Koumatos (U. of Sussex), C. Lattanzio and S. Spirito (U. of L’Aquila)), which shows propagation of $H^1$-regularity for the deformation gradient of weak solutions for semiconvex stored energies. While weak solutions still exist for initial data in $L^2$, oscillations on the deformation gradient can now persist and propagate in time.

The existence of sustained oscillations in hyperbolic-parabolic system is studied systematically via examples, for paradigm systems from viscoelasticity and for the compressible Navier-Stokes system with non-monotone pressures. In several space dimensions oscillatory examples are associated with lack of rank-one convexity of the stored energy. The subject naturally leads to the problem of deriving effective equations for the associated homogenization problems. This is in general a hard problem that can in some very simple models be addressed by ideas from the kinetic formulation for conservation laws.

Cross-diffusion systems are strongly coupled parabolic systems describing phenomena in which multiple species diffuse and interact with one another, e.g. in fluid mechanics or population dynamics. Although many methods have been developed to study existence of weak solutions to such systems, uniqueness is in general an open problem. To this degree, we study a particular cross-diffusion system, known as the Maxwell-Stefan system which describes diffusive phenomena in a multicomponent system of gases. We employ renormalized solutions and give conditions under which such solutions are unique. We, then, study the relation between weak and renormalized solutions, allowing us to identify conditions that guarantee uniqueness of weak solutions. The proof is based on an identity for the evolution of the symmetrized relative entropy. Using the method of doubling the variables we derive the identity for two renormalized solutions and use information on the spectrum of the Maxwell-Stefan matrix to estimate the symmetrized relative entropy and show uniqueness.

We will consider the task of finding a domain which minimses an energy.
The considered energy will depend on the solution of a PDE within the
domain. Such a class of problem is known as PDE constrained shape
optimisation problem. For simplicity, we consider the PDE to be
Poisson's equation with Dirichlet data. This kind of problem is well
studied in the analytical community, however a satisfying practical
algorithm has been lacking. In this talk, we will give an introduction
to shape optimisation as well as consider the analysis of a practical
algorithm for solving a prototype problem, showing its convergence and
discussing associated analytical challenges.

A long-standing topic of interest is to understand solutions of the incompressible 2D Euler equations where the vorticity of the solution stays highly concentrated around a finite number of points on some interval of time, in some sense approximating the behaviour of point vortices. There are a large class of steady states that satisfy this behaviour, and also solutions that exhibit this behaviour dynamically on finite time intervals. We exhibit solutions of 2D Euler that are genuinely dynamic, and also retain this concentration of vorticity around points for all time: a configuration approximating two vortex pairs separating at linear speed, and a configuration approximating three vortices separating like a self-similar spiral at sublinear speed. Joint work with Juan Davila, Manuel Del Pino, and Monica Musso.

I will present two results dealing with the passage to the limit in aggregation-diffusion equations where obtaining standard compactness estimates is difficult. The first result, obtained in collaboration with C. Elbar and B. Perthame, concerns the kinetic derivation of the degenerate Cahn-Hilliard equation from a certain nonlocal partial differential equation. The challenge here is that all necessary a priori estimates can only be obtained for the nonlocal quantities, providing almost no information about the limiting solution itself. We introduce a novel condition on the kernel that allows us to exploit the available nonlocal a priori estimates. The second result, obtained in collaboration with J. A. Carrillo and Y. Salmaniw, concerns the existence (and uniqueness) of solutions to aggregation-diffusion equations where the kernel is only bounded and integrable, for instance, a characteristic function of a ball or a cube. Here, we take advantage of the gradient flow structure in a novel way, utilizing the dissipation of free energy and equiintegrability to control the gradient of the solution. This second work is particularly important in ecology, where the case of a characteristic function of a cube is widely used as a toy model to study the dynamics of populations.