Prof. Dr. Mikaela Iacobelli

Research

My research focuses on problems arising from the broad context of statistical mechanics. More precisely, it concerns the analysis of partial differential equations in kinetic theory, plasma physics, mathematical physics, and nonlinear PDEs.

Kinetic equations describe the evolution of systems made of a large number of particles at a mesoscopic level. They provide an intermediate viewpoint between the Newtonian dynamics of microscopic particles and macroscopic, hydrodynamical models. A central question is to understand how these different descriptions are related, and to give a rigorous justification of the passage from one level of description to another.

A common feature of my research is the search for structures which are adapted to the problem under consideration. Depending on the model, this may mean a distance which follows the kinetic flow, a splitting of the electric or electromagnetic field, or a geometric condition which captures the behaviour at infinity. These structures often make it possible to treat singular limits, long-time dynamics, or degenerate problems which are not accessible by more direct methods.

Vlasov-type equations and quasineutral limits.

A large part of my work concerns singular limits for Vlasov-type equations. In particular, I have studied the quasineutral limit for the Vlasov-Poisson equation and for the Vlasov-Poisson equation with massless or thermalized electrons. These models arise in plasma physics and describe regimes in which the Debye length becomes very small.

From the mathematical point of view, the quasineutral limit is highly singular and may be unstable. Thus, one of the main questions is not only to prove convergence, but also to understand the stability threshold of the limiting process. This requires identifying the right stability mechanisms and topologies.

In the case of the Vlasov-Poisson equation with massless electrons, an important point is to separate the singular part of the electric field from the more regular nonlinear contribution coming from the electron density. This kind of splitting is a recurring idea and allows one to isolate the part of the field responsible for the main singular behaviour and to treat the remaining part by more regular estimates. This viewpoint has been used in various contexts, from electrostatic to electromagnetic.

Some related works:

Stability methods and kinetic Wasserstein distances.

One of the tools developed in this context is based on kinetic Wasserstein distances. The guiding idea is that classical transport distances do not always reflect the geometry of kinetic equations. In kinetic models, position and velocity play different roles, and the natural transport takes place along characteristics. It is therefore useful to build distances which incorporate this structure from the beginning.

These distances provide a natural way to compare solutions along the kinetic flow and to obtain stability estimates for Vlasov-type systems. This point of view has also led to related questions in quantum optimal transport and semiclassical analysis, in particular in connection with the Hartree equation and the Vlasov-Poisson system.

Some related works:

From particles to fluids.

Another direction concerns the relationship between the quasineutral limit and the meanfield limit, namely the derivation of mesoscopic equations of Vlasov type from the Newtonian dynamics of many particles. More generally, the goal is to understand how macroscopic fluid equations can emerge from kinetic or particle systems.

In this context, I have studied the derivation of incompressible Euler-type equations from Vlasov-Poisson dynamics. I am particularly interested in regimes in which the limiting fluid equation belongs to a natural low-regularity class. This includes the derivation of Yudovich solutions of the incompressible Euler equations from the Vlasov-Poisson system.

Some related works:

Phase mixing, Landau damping, and scattering.

I am also interested in the long-time behaviour of collisionless kinetic equations. This includes phase mixing phenomena, Landau damping, scattering, and the mechanisms which determine whether a kinetic system relaxes, oscillates, or remains close to an equilibrium.

I have studied Landau damping for the Vlasov-Poisson system with massless electrons, as well as scattering problems for Vlasov-type equations on the torus. These questions are closely related to the stability theory of plasmas and to the broader problem of understanding how mixing and oscillations affect the dynamics of collisionless systems.

Some related works:

Magnetic and electromagnetic plasma models.

More recently, I have started studying kinetic models in which magnetic and electromagnetic effects play an important role. This includes magnetized Vlasov-Poisson systems and the quasineutral limit from relativistic Vlasov-Maxwell equations to electron-MHD. These questions are at the centre of my SNSF Starting Grant Challenges and Breakthroughs in the Mathematics of Plasmas.

The Vlasov-Maxwell case requires new ideas with respect to the electrostatic Vlasov-Poisson setting. Although corrector methods were already present in the quasineutral theory of Vlasov-Poisson, the electromagnetic case contains additional oscillatory effects, coming from the magnetic field and from the solenoidal part of the electric field. A key point is to find a refined decomposition of the electromagnetic field which separates these oscillations and makes it possible to construct the appropriate dispersive correctors.

This direction is part of a broader program aimed at understanding the rigorous emergence of macroscopic plasma and fluid equations from collisionless kinetic models. It also brings together several themes which appear elsewhere in my work: singular limits, oscillations, stability estimates, and decompositions of the field adapted to the structure of the equations.

Some related works:

Steady states and analogies with fluid equations.

I have also studied the geometric structure of stationary solutions for the gravitational Vlasov-Poisson system, in the spirit of the analogy between Vlasov-type equations and two-dimensional incompressible Euler equations. These questions concern rigidity, uniqueness, and stability properties of such steady states, and more generally the way kinetic models reflect structures which are familiar in fluid mechanics.

Some related works:

Quantization of measures, Riemannian manifolds, and ultrafast diffusion.

A second line of research concerns the quantization of measures, namely the problem of approximating a diffuse measure, in an optimal way, by a finite number of points. This problem appears in several contexts, including signal processing, numerical analysis, economics, and pattern recognition.

My work in this direction has focused on two related aspects. The first is a dynamical point of view on quantization, based on gradient flows and on the degenerate parabolic equations that arise from them. This led in particular to the analysis of very fast and ultrafast diffusion equations, where questions of well-posedness, asymptotic behaviour, and concentration phenomena are closely related to the geometry of the quantization problem.

The second aspect concerns asymptotic quantization on Riemannian manifolds, including noncompact manifolds. In the non-compact case, the geometry at infinity becomes extremely relevant. Indeed, volume growth, covering properties, and the behaviour of the manifold at large scales influence whether the classical asymptotic quantization formula remains valid. This direction has developed further through covering growth estimates, which provide a more flexible way to capture the large-scale geometry of the underlying manifold.

Some related works:

Common themes.

Although these topics belong to different mathematical communities, they are connected by a common interest in the emergence of effective models from systems with many degrees of freedom. In the kinetic part, the objects are typically physical particles, such as gas molecules, ions, or electrons in a plasma. In the quantization part, one studies the optimal approximation of continuous distributions by finite configurations of points.

In both cases, the aim is to understand how discrete or microscopic structures give rise to continuum models, and how the geometry of the problem influences the limiting behaviour. Across these different problems, the approach is often to identify the mathematical structure which is hidden in the model, from the correct metric to the correct decomposition or geometric condition.