Marco Bresciani (Johannes Kepler University Linz, Austria)
Title: Variational models for nematic elastomers at large strains
Abstract: We study the nonlinear variational model for nematic elastomers proposed by Barchiesi and DeSimone, in which the director field is defined on the unknown deformed configuration. After introducing the appropriate functional framework, we establish the existence of minimizers in the static setting and construct quasi-static evolutions in the rate-independent regime. Finally, we extend the model to a class of deformations allowing cavitation. The talk is based on joint work with Elisa Davoli (TU Wien), Manuel Friedrich (JKU Linz), Martin Kružík (Czech Academy of Sciences, Prague), Carlos Mora-Corral (Autonomous University Madrid), and Bianca Stroffolini (University of Naples).
José A. Carrillo (University of Oxford, United Kingdom)
Title: Mean-Field Derivation of the Space-Homogeneous Landau Equation via BBGKY Hierarchy Method
Abstract: We consider the Kac particle system for the space-homogeneous Landau equation. For the Coulomb potential, we show that the Fisher information of the Liouville equation is monotonically decreasing in time. The monotonicity ensures the compactness to derive a weak solution of the Landau hierarchy. This is a work in collaboration with Shuchen Guo. For hard potentials, Shuchen Guo has shown that the propagation of exponential moments holds at the particle level. These moments bound ensures the uniqueness of weak solutions of the Landau hierarchy, which implies propagation of chaos for this range.
Patrick E. Farrell (University of Oxford, United Kingdom)
Title: Analysis of a compressible Oseen-Frank energy
Abstract: We study a compressible variant of the one-constant Oseen--Frank energy for nematic liquid crystals in which the elastic energy density is weighted by a prescribed thermodynamic pressure field. This model arises from a kinetic derivation based on the Boltzmann--Curtiss equation and provides a natural mechanism for regularizing defects through the possible degeneracy of the pressure. Motivated by this structure, we formulate the variational problem in Muckenhoupt weighted Sobolev spaces and establish an existence theory for minimizers under prescribed Dirichlet data. Numerical experiments illustrate the theoretical results.
Maria Groppi (University of Parma)
Title: Kinetic Boltzmann-BGK modeling of inert and reactive mixtures
Abstract: In classical kinetic theory, the dynamics of gas mixtures is naturally described by integro-differential Boltzmann equations for the species distribution functions. However, the analytical and numerical study of these equations is rather difficult due to the complexity of the nonlinear integral Boltzmann operator, which describes collision mechanisms in detail. For this reason, starting from the BGK model for a single gas proposed by Bhatnagar, Gross, and Krook in 1954, several relaxation models for mixtures have been introduced. In particular, in this talk we start from a consistent BGK model for inert mixtures proposed in 2018 by Bobylev et al., characterized by a collision operator for each species given by a sum of bi-species BGK operators, and present two extensions. The first one is a hybrid Boltzmann-BGK model for an inert mixture of monatomic gases, which combines the detailed description of collisions provided by the Boltzmann integral operators with the simplicity and numerical tractability of BGK-type relaxation operators. This kinetic model has the same structure as the full Boltzmann equations, with the collision term of each constituent given by a sum of bi-species operators, which may be chosen to be either of Boltzmann or BGK type. The second one is a BGK-type model for a chemically reactive mixture of four gases, whose collision operator for each constituent is expressed as the sum of two terms: a sum of binary BGK operators accounting for elastic mechanical interactions, and a single BGK-type term describing a bimolecular reversible chemical reaction. The BGK relaxation operators involve suitable Maxwellian attractors depending on auxiliary parameters, which must be properly computed in order to make the BGK model consistent. This is done by requiring the exchange rates of momentum and energy for the BGK and Boltzmann operators to coincide, under the assumptions of Maxwell molecule mechanical potentials and slow chemical reactions. For both models, we prove that the conservation laws and equilibria are correctly reproduced, and investigate the hydrodynamic limits.
Kaibo Hu (University of Oxford, United Kindgom)
Title: Long-Time Asymptotics and Discrete Antidynamo Theorems for Differential Forms
Abstract: Advection–diffusion of differential forms provides a unified framework for several models in mathematical physics, including scalar transport, Fokker–Planck equations, and kinematic magnetohydrodynamics. While most numerical analyses focus on convergence over finite time intervals, many physically relevant questions concern the behavior of a discretization at fixed mesh size as time tends to infinity. In particular, a numerical method should not produce spurious dynamo growth when the continuous problem satisfies an antidynamo principle. In this talk, we will present a finite element exterior calculus framework for the long-time analysis of Lie advection–diffusion. For potential flows, the operator can be reformulated as a weighted Hodge Laplacian. This yields a weighted Hodge decomposition, identifies the stationary space with the de Rham cohomology, and proves exponential convergence to a weighted harmonic form. The corresponding weighted finite element discretization preserves these properties for every mesh size. For general advection fields, we introduce a compatible mixed finite element scheme and show that the space of closed discrete stationary k-forms has dimension at least the k-th Betti number. Under suitable spectral-preservation assumptions, the discrete dynamics converges exponentially to stationary states. In two dimensions, every discrete one-form converges to a closed stationary form for any positive diffusion coefficient, thereby excluding spurious discrete dynamo modes. These results illustrate how FEEC can preserve not only local differential structure, but also topology-driven long-time dynamics.
Josef Málek (Charles University, Czech Republic)
Title: Rate-type viscoelastic fluids without or with stress diffusion: modeling, analysis, and rod-climbing simulations
Abstract: Maxwell, Oldroyd, and Burgers viscoelastic models are examples of a class of incompressible non-Newtonian fluids known as ``rate-type viscoelastic fluids". In these models, the development of part of the Cauchy stress is governed by a first-order or second-order tensorial evolutionary equation. In their classical form, these equations do not include stress diffusion. Stress diffusion can be added in order to capture interesting physical phenomena, such as finite-thickness structures in shear banding. In the talk, we first provide an unified thermodynamic basis on these models. Then we will report on the long-time and large-date existence theory for an internal flows of such fluid in a bounded three-dimensional domains. Finally, we demonstrate the performance of selected viscoelastic models in the computational reproduction of experimental data for rod climbing, i.e., the rise of a fluid along a rod rotating about its axis, an effect associated with the non-Newtonian phenomenon known as normal-stress differences.
Peter Markowich (KAUST, Saudi Arabia)
Title: Continuum Modelling of Biological Network Emergence, Formation and Evolution
Abstract: This talk will give an overview of recent results for partial differential equation models for the emergence, formation and evolution of biological transport networks. Typically, the models describe the material pressure field using a Darcy's type equation and the dynamics of the network conductance. Randomness in the porous medium material structure is represented by linear diffusion and conductance relaxation by an algebraic decay term. Micro- and mesoscopic models will be introduced and will show how they are connected to the classical macroscopic PDE system (by D. Hu and D. Cai) and to discrete graph based models.
Carmela Moschella (University of Oxford, United Kingdom)
Title: Continuum Models for Hard Anisotropic Particles
Abstract: Volume-exclusion interactions play a crucial role in the self-organisation observed in many physical and biological systems. In particular, they are fundamental to explain the spontaneous emergence of nematic order in populations of anisotropic particles, as observed, for example, in liquid crystals and dense suspensions of myxobacteria. In this talk, we consider a stochastic model of hard-core anisotropic particles in two dimensions, represented as non-overlapping rectangular particles undergoing Brownian motion with drift. Starting from this particle model, we use matched asymptotic expansions in the dilute regime to derive a kinetic equation for the one-particle probability density in position and orientation space. The resulting kinetic model explicitly shows how excluded-volume effects are encoded in the interaction kernel, which depends on particle aspect ratio. In the limit of vanishing width, we recover the corresponding description for non-overlapping needles. We perform a linear stability analysis of the system in the spatially homogeneous setting, which reveals an isotropic-to-nematic transition at increasing density, in agreement with Onsager's theory. We then extend the analysis to self-propelled particles and recast the continuum equation in a form directly analogous to that obtained for active Brownian hard spheres. This comparison shows that shape anisotropy systematically suppresses motility-induced phase separation, which is instead present in the isotropic case. Finally, we discuss possible extensions to ellipsoidal or capsule-shaped particles.
Lorenzo Pareschi (Heriot-Watt University, Edimburgh, United Kingdom)
Title: Collective learning across scales
Abstract: Many modern optimization and artificial-intelligence methods rely on populations of interacting agents evolving on different time scales. A fast dynamics updates model parameters or particle states, while a slower dynamics adapts hyperparameters, strategies, or interaction rules. This structure appears naturally in population-based training of neural networks and in adaptive swarm-based optimization. In this talk, I will discuss a kinetic perspective on such multiscale learning systems. Starting from interacting-particle descriptions, I will show how mean-field equations provide a statistical representation of populations evolving jointly in state and strategy spaces. When the fast dynamics relaxes sufficiently rapidly, its equilibrium or effective statistical state determines the quantities driving the slower adaptation. This leads to reduced equations related to selection–mutation dynamics.
Giovanni Russo (University of Catania, Italy)
Title: Dispersive behaviour of quasilinear hyperbolic waves on periodic background
Abstract: Waves propagating in media with periodic structures have attracted a lot of attention in recent years. Such media show interesting macroscopic properties, which may be quite different from those of the individual materials constituting the stratified system. More recently, waves on fluids and gases with an underlying periodic structure have been studied. These waves present a peculiar, somehow unexpected, behaviour. For example, there is evidence that in spite of the fact that the waves are governed by genuinely quasi-linear hyperbolic system, they do not break into a shock. Here we present several systems in which hyperbolic waves show a dispersive behaviour, and deduce approximate effective equations which explain the peculiar phenomena. In all cases, the effective equations are obtained by asymptotic expansion of the solution in a small parameter representing the ratio between the period of the structure and a typical wavelength. We start considering 1D shallow water system with periodic bathymetry. The detailed numerical solution of the system is compared with effective equations at various orders in the small parameter. Traveling wave solutions of the dispersive models are also computed and compared with the traveling waves emerging from the shallow water system. As a second example we consider 2D shallow water \cite{DispersiveSW1D}, with waves propagating in the $x$-direction, while the bathymetry is periodic in the $y$ direction. The last example concerns the Euler equations in gas dynamics. The stationary background is a state with constant pressure, zero velocity, and a periodic variation in the density. Numerical experiments on multilayer stiffened gas show a similar behaviour.
Thomas Surowiec (SIMULA, Oslo, Norway)
Title: The Latent Variable Proximal Point Method: A New Solver Paradigm for Variational Inequalities, Nonlinear PDEs, and Beyond
Abstract: The Latent Variable Proximal Point (LVPP) method is a novel, geometry‐encoding scheme in which the continuous level informs the algorithms, discretization techniques, and implementation. Mathematically speaking, it embeds the problem at hand into a sequence of related saddle‐point problems by introducing a structure‐preserving transformation between a latent Banach space and the feasible set. LVPP arises at the confluence of information geometry, optimization, and convex analysis through its use of proximal point methods, Legendre functions, and the isomorphisms induced by their gradients. The method yields algorithms with mesh‐independent convergence behaviour for obstacle problems, contact, topology optimization, fracture, plasticity, and more; in many cases, for the first time.
Tim van Beeck (University of Göttingen, Germany)
Title: Tuning Sound with Liquid Crystals: Finite Element Methods for Nematoacoustics
Abstract: The propagation of acoustic waves in nematic liquid crystals is influenced by the order structure of the fluid. The nematic director renders both the attenuation and the speed of sound anisotropic, so that sound travels fastest along the direction of molecular alignment. Because the director can itself be reoriented by an external electromagnetic field, this anisotropy is tunable and the propagation of a wave can be controled from the outside. In the frequency domain, these effects can be modeled by the nematic Helmholtz--Korteweg (NHK) equation. It augments the classical Helmholtz equation with two fourth-order terms, one of which is anisotropic along the nematic director. This anisotropic higher-order term produces the desired directional effects, yet it is also the main difficulty in the mathematical analysis and the numerical discretisation. In this talk, we first study the well-posedness of the weak formulation of the NHK equation. We then turn to its discretisation by $H^2$-conforming and non-conforming $C^0$-interior penalty finite element methods, and outline the main ingredients of their stability and convergence analysis. Numerical experiments in two and three dimensions illustrate the anisotropic effect of the nematic director, with applications to acoustic scattering.
Niccolò Tassi (University of Granada, Spain)
Title: Self-similarity and diffusive limits for linear kinetic equations
Abstract: In this talk, I will present some recent results obtained with José Cañizo and Stéphane Mischler regarding the asymptotic behaviour of some linear kinetic equations without confinement. We present the results for three types of operators and their fractional variants. The unconfined setting prevents the use of standard hypocoercivity arguments and, to the best of our knowledge, the decay of the L^2 norm of the solution in an unconfined setting was only proved recently. We improve these results by providing a more detailed description of the asymptotics, not only establishing norm decay but also identifying the exact asymptotic profile of the solution. After rescaling, this allows us to show a mesoscopic-to-macroscopic limit uniformly in time, which is also new. At the end of the talk, I will briefly present ongoing extensions of these results to more general collision operators.
Umberto Zerbinati
Title: TBA
Abstract: TBA