Partial Differential Equations (PDEs) section
Invited speakers:
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Programme (all talks are in Abacws/3.38):
Mon 30th March, 2026
| 3:00pm – 3:30pm | Nikos Katzourakis (Reading) |
| “A measure-theoretic generalisation of Danskin’s theorem and everywhere semi-differentiability of \(L^\infty\) functionals” (abstract) | |
| 3:30pm – 4:00pm | Paolo Salani (Florence) |
| “Concavity and convexity versus heat flow” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 5:00pm | Zhengyan Wu (Munich) |
| “Fluctuating hydrodynamics in kinetic equations” (abstract) | |
| 5:00pm – 5:30pm | Prachi Sahjwani (Cardiff) |
| “Stability of Quermassintegral and Minkowski-Type Inequalities in Curved Spaces” (abstract) |
Tue 31st March, 2026
| 3:00pm – 3:30pm | Elaine Crooks (Swansea) |
| “Self-similar fast-reaction limits of reaction-diffusion systems with nonlinear diffusion” (abstract) | |
| 3:30pm – 4:00pm | Josephine Evans (Warwick) |
| “Kinetic equations which converge to the porus medium and fast diffusion equations” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 5:00pm | Megan Griffin-Pickering (Zurich) |
| 5:00pm – 5:30pm | Amit Einav (Durham) |
| “Can we have a little order in all this chaos?” (abstract) |
Wed 1st April, 2026
| 3:00pm – 3:30pm | Havva Yoldaş (TU Delft) |
| “Fisher information and the well-posedness of the Landau equation” (abstract) | |
| 3:00pm – 4:00pm | Karsten Matthies (Bath) |
| “Fronts in dissipative Fermi-Pasta-Ulam-Tsingou chain” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 5:00pm | Francesco Mainardi (Bologna) |
| “Transient waves in linear dispersive media with dissipation” (abstract) |
Partial Differential Equations (PDEs) section organisers:
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Abstracts:
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Elaine Crooks (Swansea): Self-similar fast-reaction limits of reaction-diffusion systems with nonlinear diffusion
This talk is concerned with the characterisation of fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. The ideas used extend previous results in the linear diffusion case and show that in the fast-reaction limit, spatial segregation leads to the two components of the original systems each converging to the positive and negative parts of a self-similar limit profile that satisfies one of four ordinary-differential systems. The position of the free boundary separating where such self-similar profiles are positive from where they are negative provides information on the rate of penetration of one substance into the other and for specific forms of nonlinear diffusion, some results will be presented on the relationship between the form of the nonlinear diffusion and the position of this free boundary. This is joint work with Yini Du.
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Amit Einav (Durham): Can we have a little order in all this chaos?
Systems that revolve around the interactions of many elements are a constant part of our day to day lives. Yet for all their prevalence, trying to explore mathematical models of such systems, theoretically or numerically, can often be a herculean task. To try and address this difficulty, it was realised early on (back in the late 19th century) that we do not need to understand how each and every element in the system behaves. Instead, it is often enough to understand how a typical or average element does.
A revolutionary idea that birthed a new way to investigate systems of many elements was formalised in the work of Mark Kac in 1956. Kac suggested to find how an average particle in dilute gas behaves by considering a “probabilistic model” of the gas, expressed via a PDE for the probability to find the system in various configurations, together with the idea that as the number of particles in the gas increases, they become more and more independent. The latter is often known as molecular chaos, or chaos. Combining these two ingredients, Kac was able to find an equation that describes how a “limiting average particle” evolves in his settings – which ended up being a one-dimensional variant of the celebrated Boltzmann equation.
These ideas are far more general and powerful than their application in Kac’s model, and they have formed the framework of what we now call the mean field limit approach. At its heart, the mean field limit approach has two ingredients:
- An average model for the system, expressed via a PDE for its probability measure.
- An asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity.
In recent decades the use of the mean field limit approach has expanded beyond physical models into the realm of biology, economy, and societal studies. Yet in almost all cases considered so far, the sole correlation relation used was chaos. This seems to be inappropriate in settings that have a tendency for adherence such as biological swarming.
In our talk we will discuss the background to the mean field limit approach as well as Kac’s model and its limiting equation. We will then jump to 2013 and consider the Choose the Leader model, or CL model, which is a Kac-like animal swarming model introduced by Carlen, Degond, and Wennberg where chaoticity breaks. Motivated by the desire to understand this model better, we will introduce two new asymptotic correlation relations, order and partial order, and see how they arise naturally in the CL model.
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Josephine Evans (Warwick): Kinetic equations which converge to the porus medium and fast diffusion equations
This talk is based on a joint work with Daniel Morris and Havva Yoldas. I will discuss a class of nonlinear kinetic equations which converge to nonlinear diffusion equations. I will explain how we have shown well-posedness, long time behaviour and convergence to a diffusive limit. I will discuss how these equations are related to more physically relevant equations and interesting future research direction.
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Nikos Katzourakis (Reading): A measure-theoretic generalisation of Danskin’s theorem and everywhere semi-differentiability of \(L^\infty\) functionals
J. Danskin proved in the 1960s that the envelope of functions \(F : \mathbb R^n \times K \to \mathbb R\), where \(K\) is compact set in \(\mathbb R^m\), given by \[ f(u) := \max_{k \in K} F(u,k), \] Is (1-sided directional) semi-differentiable everywhere on its domain of definition, under continuity assumptions in the second variable and differentiability in the first. This result has paramount importance for numerous applications, including game theory (JD was working as a US Navy researcher during the Cold War on battle strategies), economics, finance, stochastic optimisation, portfolio selection, etc, and relates to the so-called envelope theorem in comparative statics. Due to its significance, various extension have been proved since, but none of them allows to replace the maximum over a compact set with its "essential" measure-theoretic counterpart.
In this talk I will expound on a most recent development, allowing to generalise this result to functions defined the product of a Banach space with a measure space, utilising some new measure-theoretic machinery. As a direct consequence, we obtain the previously unknown regularity property that all supremal functionals in the Calculus of Variations in \(L^\infty\) are semi-differentiable everywhere, with an explicitly known semi-differential. Given these are generally nowhere Gateaux differentiable, semi-differentials serve as an intrinsic substitute of the Euler-Lagrange equations.
As a consequence, we derive an intrinsic variational characterisation for supremal functionals via PDEs, which on the one hand renders the need for \(L^p\) approximations obsolete, and on the other hand allows to derive the usual Aronsson-type equations and systems, the latter being only necessary condition in general.
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Francesco Mainardi (Bologna): Transient waves in linear dispersive media with dissipation
In the study of linear dispersive media it is of primary interest to gain knowledge of the impulse response of the material. The standard approach to compute the response involves a Laplace transform inversion, i.e., the solution of a Bromwich integral, which can be a notoriously troublesome problem. In this talk we propose a novel approach to the calculation of the impulse response, based on the well assessed method of the steepest descent path, which results in the replacement of the Bromwich integral with a real line integral along the steepest descent path. In this exploratory investigation, the method is explained and applied to the case study of the Klein- Gordon equation with dissipation, for which analytical solutions of the Bromwich integral are available, as to compare the numerical solutions obtained by the newly proposed method to exact ones. Since the newly proposed method, at its core, consists in replacing a Laplace transform inverse with a potentially much less demanding real line integral, the method presented here could be of general interest in the study of linear dispersive waves in presence of dissipation, as well as in other fields in which Laplace transform inversion come into play. Joint work with Andrea Mentrelli (University of Bologna) and Juan Luis González-Santander (University of Oviedo).
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Karsten Matthies (Bath): Fronts in dissipative Fermi-Pasta-Ulam-Tsingou chain
We consider a spatially discrete version of a damped nonlinear wave equation. In a dissipative Fermi-Pasta-Ulam-Tsingou chain particles interact with their nearest neighbors through nonlinear potentials (e.g. Hertzian) and linear dissipative forces. We show the existence of front solutions connecting two different uniformly compressed (or stretched) states at $\pm \infty$ using an implicit function argument starting at a suitable continuum limit in the case of large damping. The main technical difficulties arise when showing continuity of the relevant derivatives for a setting that includes Hertzian potentials requiring tools for exponentially weighted, fractional Sobolev spaces including the Kato-Ponce inequality. A more detailed analysis allows us to determine sharp exponential decay rates which imply monotonicity of the waves. We identify two different auxiliary continuum limits near the two asymptotic states of the front, these will yield the correct decay rates by a careful analysis of their Fourier symbols. Solutions can be shown to have these exact rates by applying a conditional improvement lemma for a finite number of times. Joint work with Michael Herrmann and Guillaume James.
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Prachi Sahjwani (Cardiff): Stability of Quermassintegral and Minkowski-Type Inequalities in Curved Spaces
Geometric inequalities lie at the heart of convex geometry and geometric analysis. A classical example is the isoperimetric inequality, and in higher dimensions this generalises to the quermassintegral inequalities, which relate curvature integrals of a convex domain to one another. Once a sharp inequality is known, a natural question arises: if a domain nearly achieves equality, must it be close in shape to a geodesic sphere? This is the question of stability. A quantitative answer showing that the deficit controls the geometric distance to a sphere gives far more information than the inequality alone. In Euclidean space this is well-understood, but in curved spaces such as hyperbolic space the problem is much harder, and until recently no stability results were known despite the sharp inequalities having been established.
In this talk, I will present new quantitative stability results in two settings. The first is hyperbolic space, where we prove stability of the quermassintegral inequalities for horo convex hypersurfaces. The second is warped product manifolds where we establish stability of Minkowski-type inequalities. In both cases, the conclusion is that a small deficit forces the hypersurface to be close to a geodesic sphere, with the distance controlled by the Lp-norm of the traceless second fundamental form. The proofs combine curvature flow methods with quantitative rigidity arguments, adapting ideas from the Euclidean setting to these non-Euclidean spaces.
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Paolo Salani (Florence): Concavity and convexity versus heat flow
There is a strong connection between log-concavity and the heat flow. In particular, it is well known that the heat flow preserves the log-concavity of the initial datuma and that log-concavity spontaneously and eventually appears whenever starting with a compactly supported initial datum. Together with K. Ishige and A. Takatsu, we deeply investigated the preservation of concavity properties by the heat flow and, recently, together with K. Ishige and T. Petitt, we also pointed our attention to convexity properties. I will try to give a picture of the state of the art.
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Zhengyan Wu (Munich): Fluctuating hydrodynamics in kinetic equations
We present a recent well-posedness result for the Vlasov-Fokker-Planck-Dean-Kawasaki (VFPDK) equation with correlated noise. This stochastic kinetic equation arises as a fluctuating mean-field limit of second-order Newtonian particle systems, bridging classical kinetic theory and stochastic fluctuation effects. The model features bounded nonlocal interactions together with a diffusion coefficient of square-root type, which introduces significant analytical difficulties. To overcome these issues, the analysis combines refined kinetic semigroup estimates with the framework of renormalized kinetic solutions.
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Havva Yoldaş (TU Delft): Fisher information and the well-posedness of the Landau equation
I will report on the breakthrough result by Guillen and Silvestre (Acta Math, 2025) where they show the global well-posedness of the spatially homogeneous Landau-Coulomb equation. The Landau-Coulomb equation is a kinetic equation describing the statistical evolution of a plasma dominated by long-range Coulomb interactions. The result is based on proving that the Fisher information is monotonically decreasing along the solutions of this equation. They do so by considering a "lifted equation" where the dimension of the velocity space is doubled. In this lifted regime, the nonlinear degenerate Landau operator turns into a linear operator. They then prove the inequalities relating the Fisher information along the Landau operator and the Fisher information along the lifted operator which is studied by a special change of variables. Their result crucially relies on the symmetry assumption for the joint distribution of two colliding particles. I will finish by summarising our recent results obtained in a collaboration with J. Junné (TU Delft) and R. Winter (Cardiff) on the multi-species Landau equation where we construct more general Lyapunov functionals removing this symmetry requirement.
Contact:
All further questions and inquiries can be addressed to:
- DragoniF at cardiff dot ac dot uk
- WinterR at cardiff dot ac dot uk