Kinetic theory describes gases and plasmas as large collections of interacting molecules, from a statistical point of view. Its most famous success, due to Boltzmann, is the first analytic derivation of the monotonicity in time of the entropy.
In most gases of practical interest, e.g. N2, O2 or CO2, molecules are polyatomic: composed of multiple atoms. This geometrical structure induces rotational and vibrational effects, which are a challenge to include in the theory. We present a general framework for modeling polyatomic molecules in kinetic theory, based on elementary physical ideas and formulated in a measure-theoretic setting. We show that within this framework one may adopt different perspectives, each relevant for a different kind of problem (physical modeling, mathematical analysis or numerical simulation), clarifying and unifying the previous approaches in the literature.
We consider determinantal point processes $\Lambda_\phi$ arising from generalized Fock spaces $\mathcal F_\phi$, defined by a doubling subharmonicweight $\phi$. We provide a complete characterization of when these processes are almost surely separated with respect to theeuclidean metric on the plane. As a consequence, we also obtain a characterization of when these processes are almostsurely interpolating for the classical Fock spaces. Furthermore, we emphasize the role of intrinsic repulsion in determinantalprocesses by comparing $\Lambda_\phi$ with the Poisson process with the same first intensity. Additionally, we demonstrated that, withprobability one, the point process $\Lambda_\phi$ fails to be separated with respect to the distance induced by the reproducing kernel$K_\phi$.
In this seminar, we will explore a fractional version of a semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980s. The problem arises when considering steady states of the Keller–Segel model with nonlocal diffusion of chemical concentration. In the fractional setting, there are different possible notions of 'nonlocal Neumann conditions'. We will study the system with both spectral conditions, following Stinga and Volzone, and integral conditions (introduced by Dipierro, Ros-Oton, and Valdinoci), as investigated by Cinti and Colasuonno. While an extension theorem analogous to that of Caffarelli and Silvestre exists for the spectral boundary condition, no such theorem is available for integral boundary conditions. This makes it more challenging to obtain Liouville-type theorems and to prove the nonexistence of non-trivial positive solutions when the concentration parameter is sufficiently large.
Based on joint projects with Eleonora Cinti (UNIBO), Marco Ghimenti (UNIPI), Juncheng Wei (CUHK), and Tobias Weth (GUF).
In this talk, we consider partial differential equations of parabolic $p$-Laplace type in noncylindrical domains $U = \bigcup_{t \in [0,T]} \{t\} \times \Omega_t$, where the underlying spatial set $\Omega_t \subset \mathbb{R}^n$ may change in time. Our goal is to characterize time-dependent boundary values that yield the existence of a solution, focusing on domains $U$ that satisfy a Lipschitz condition with respect to a parabolic metric. In particular, we discuss the extension to $U$ of certain functions that are only defined on the parabolic boundary of $U$, and conversely the trace space for functions defined on $U$ and contained in a suitable parabolic function space. The talk is based on joint work with Daniel Campbell and Sebastian Schwarzacher.
