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For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with bubbling sources. If the strength of the bubbling sources at blowup points are not multiple of $4\pi$ we prove that bubbling solutions are unique under non-degeneracy assumptions. This work extends a previous work of Bartolucci, et, al \cite{bart-4}., Comment: 36 pages

A hierarchy of models for pedestrian flow is numerically investigated using particle methods. It includes microscopic models based on interacting particle system coupled to an eikonal equation, hydrodynamic models using equations for density and mean velocity, nonlocal continuum equations for the density and diffusive Hughes equations. Particle methods are used on all levels of the hierarchy. Numerical test cases are investigated by comparing the above models.

We consider the fractional mean-field equation on the interval [Formula: see text] [Formula: see text] subject to Dirichlet boundary conditions, and prove that existence holds if and only if [Formula: see text]. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of nonlocal type.

This paper is concerned with the mean field equation of the equilibrium turbulence on a compact Riemannian surface. The solution existence is shown in the super-critical case without any assumption on the topology and the geometry of the surface.

The authors investigate the stability of a steady ideal plane flow in an arbitrary domain in terms of the L2 norm of the vorticity. Linear stability implies nonlinear instability provided the growth rate of the linearized system exceeds the Liapunov exponent of the flow. In contrast, a maximizer of the entropy subject to constant energy and mass is stable. This implies the stability of certain solutions of the mean field equation.

We extend a mapping from infinite dimensional to zero dimensional (impurity) models to derive mean field equations of several strongly correlated systems which become exact in infinite dimensions. We discuss various magnetic phases of the Hubbard model, the periodic Anderson model the Kondo lattice and the Falicov Kimball model and we relate them to different impurity models. Qualitative insights into these models is gained from the exact results obtained for their zero dimensional counterparts.

Using a new method [19], it is possible to derive mean field equations from the microscopic N body Schrödinger evolution of interacting particles without using BBGKY hierarchies. This method also allows for error estimates and can be generalized to systems with external fields, of which both points are relevant from a physics perspective. Recently, this method was used to derive the Hartree equation for singular interactions [11] and the Gross–Pitaevskii equation without positivity condition on the interaction [17] where one had to restrict the scaling behavior of the interaction. Assuming positivity of the interaction, this paper deals with more general scalings including the so-called Gross–Pitaevskii scaling.