Analytical, numerical, and applied perspectives on diffusion
Three main lecture series (8 hours each) and two invited junior one-hour seminars.
Focus on reaction-diffusion systems, structured population dynamics, and coherent structures.
You can reach us at: naplespir2026.diffphen@gmail.com
Deadline: April 30, 2026
Please fill out the following registration form for the Summer School "Diffusive Phenomena"
Registration form
We introduce the Shigesada-Kawasaki-Teramoto (SKT) system of population dynamics, as a macroscopic system of 2 cross diffusion equations for 2 spatially-structured species in competition. We explain how this system can be formally obtained as a fast reaction limit of a more standard reaction-diffusion system of three equations. We then present the linear stability estimate showing the appearance of Turing instability for the SKT system, and discuss its consequences. We establish the basic a priori estimates for this system in different situations (in the so-called triangular and non-triangular cases), and explain how they can be used in order to show the existence of weak solutions to the system. We also introduce the duality method allowing to show that those solutions are strong in dimension (one and) two. Finally, we present the ideas of a recent work allowing to get the same result in dimension three.
Reaction–diffusion systems may exhibit diffusion-driven
instabilities leading to the formation of spatial patterns.
While linear stability analysis detects the onset of a Turing
instability, understanding the properties of the emerging
patterns requires a nonlinear analysis of the dynamics near
the bifurcation point. These lectures focus on the behavior of
reaction-diffusion systems close to a Turing bifurcation. We
present Lyapunov-Schmidt reduction and weakly nonlinear
analysis as analytical tools for studying the dynamics near
the instability threshold. In this framework one derives
amplitude equations that describe the slow evolution of
pattern amplitudes and provide information on pattern
selection and nonlinear interactions between unstable
modes.
We discuss how these reduced equations relate to
the original reaction-diffusion system and examine the range
of validity of the amplitude equation formalism. The analysis
also includes secondary instabilities such as the Eckhaus
instability, which determines the stability of spatially
periodic patterns. The aim of the lectures is to introduce
analytical techniques for investigating nonlinear pattern
formation in diffusion-driven systems and to show how reduced
descriptions capture the mechanisms governing pattern
selection near the Turing threshold.
Reaction-diffusion systems provide a fundamental framework for spontaneous pattern formation in spatially extended systems. A key mechanism is diffusion-driven instability, first proposed by Alan Turing, through which diffusion can destabilize a spatially homogeneous equilibrium and lead to the emergence of stationary spatial patterns. These lectures introduce the mathematical foundations of diffusion-driven instability and the emergence of Turing patterns. Starting from general reaction-diffusion models, we explore the interplay between reaction kinetics and diffusion, presenting linear stability analysis as a tool to detect the onset of instabilities and identify pattern-forming modes. Its advantages and limitations are highlighted, motivating the need for further nonlinear analysis . Classical models will be discussed in chemical and biological contexts and, through applied examples, we also examine the impact of initial conditions, domain geometry and growth mechanisms on pattern formation and selection. The aim of the course is to provide an introduction to the mathematical mechanisms underlying Turing pattern formation and to illustrate how theoretical analysis, modeling and experimental evidence together contribute to our understanding of diffusion-driven self-organization in spatially extended systems.
Modeling physical phenomena on curved surfaces, from simple heat flow to more complex gravity-driven geophysical processes, transport in geological formations, or dynamics on biological membranes, naturally leads to surface PDEs where geometry fundamentally influences the underlying processes. Capturing this geometry-physics interplay requires intrinsic formulations: mathematical frameworks that reformulate three-dimensional surface problems as genuinely two-dimensional equations, maintaining a 2D computational setting while naturally incorporating geometric effects. Beyond modeling advantages, intrinsic formulations extend naturally to numerical discretization. Geometry-adapted schemes inherit the intrinsic structure while automatically ensuring that solutions remain tangent to and constrained on the surface, avoiding embedding artifacts. In this talk, I will discuss the development of intrinsic discretization techniques, and show their effectiveness through application in hydraulics, hydrology, and biophysics.
Diffusive systems describe relevant phenomena across a wide
variety of fields, ranging from ecology and biology to
chemistry and medicine. In such systems, spatial patterns and
localised structures often emerge as forms of
self-organisation resulting from activator-inhibitor dynamics.
An important example of this type is given by excitable media,
where the activator reacts rapidly to external stimuli and the
inhibitor regulates and limits the activator dynamics, thus
creating a balance in the system. Among these coherent
structures, travelling waves play a particularly prominent
role, as they provide a fundamental mechanism for the spatial
propagation of activity, signals, or densities through
space.
Understanding the evolution of such solutions, as
well as identifying the mechanisms that determine their main
features—such as structure and propagation speed—is therefore
of significant interest. In this context, it is particularly
useful not only to prove the existence of travelling waves,
but also to infer their qualitative and quantitative
properties. Geometric Singular Perturbation Theory (GSPT)
provides a powerful framework for this purpose: it allows for
a constructive proof of existence of travelling wave solutions
in multiscale diffusive systems and yields asymptotic
estimates for their propagation speed at leading order.
In
this talk, we investigate travelling pulse solutions in the
Barkley model, a prototypical excitable system exhibiting
activator–inhibitor dynamics. The intrinsic multiscale
structure of the model enables the application of GSPT to
construct such pulses as homoclinic orbits in the associated
three-dimensional travelling-wave phase space. The analytical
results are complemented by a detailed numerical
investigation, including direct simulations of the partial
differential equation and numerical continuation of the
travelling-wave solutions using the software AUTO.