This paper represents the first attempt to develop a theory for linear-quadratic mean field games in possibly infinite dimensional Hilbert spaces. As a starting point, we study the case, considered in most finite dimensional contributions on the topic, where the dependence on the distribution enters just in the objective functional through the mean. This feature allows, similarly to the finite dimensional case, to reduce the usual mean field game system to a Riccati equation and a forward backward coupled system of abstract evolution equations. Such system is completely new in infinite dimension and no results have been proved on it so far. We show existence and uniqueness of solutions for such system, applying a delicate approximation procedure. We apply the results to a production output planning problem with delay in the control variable.

We consider the time-harmonic Maxwell equations set on a domain made up of two subdomains that represent a magnetic conductor and a non-magnetic material, and we assume that the relative magnetic permeability $μ_{r}$ between the two materials is high. We prove uniform a priori estimates for Maxwell solutions when the interface between the two subdomains is supposed to be Lipschitz. Assuming smoothness for the interface between the subdomains, we prove also that the magnetic field possesses a multiscale expansion in powers of $1/ \sqrt{μ_{r}}$ with profiles rapidly decaying inside the magnetic conductor.

The Vicsek-Bhatnagar–Gross–Krook (BGK) equation is a kinetic model for alignment of particles moving with constant speed between stochastic reorientation events with sampling from a von Mises distribution. The spatially homogeneous model shows a steady state bifurcation with exchange of stability. The main result of this work is an extension of the bifurcation result to the spatially inhomogeneous problem under the additional assumption of a sufficiently large Knudsen number. The mathematical core is the proof of linearized stability, which employs a new hypocoercivity approach based on Laplace–Fourier transformation. The bifurcation result includes global existence of smooth solutions for close-to-equilibrium initial data. For large data, smooth solutions might blow up in finite time, whereas weak solutions with bounded Boltzmann entropy are shown to exist globally.