Abstract We investigate various versions of multi-dimensional systems involving many species, modeling aggregation phenomena through nonlocal interaction terms. We establish a rigorous connection between kinetic and macroscopic descriptions by considering the small inertia limit at the kinetic level. The results are proved either under smoothness assumptions on all interaction kernels or under singular assumptions for self-interaction potentials. Utilizing different techniques in the two cases, we demonstrate the existence of a solution to the kinetic system, provide uniform estimates with respect to the inertia parameter, and show convergence toward the corresponding macroscopic system as the inertia approaches zero.

Fractional dissipation is a powerful tool to study nonlocal physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021), a new approach is proposed to deal with dissipative systems including fractionally damped systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. The aim of this work is to derive higher-order fractional variational integrators by means of convolution quadrature (CQ) based on backward difference formulas. We then provide numerical methods that are of order 2 improving a previous result in Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021). The convergence properties of the fractional variational integrators and saturation effects due to the approximation of the fractional derivatives by CQ are studied numerically.

In this paper, we present a novel approach to the geometric formulation of solid and fluid mechanics within the port-Hamiltonian framework, which extends the standard Hamiltonian formulation to non-conservative and open dynamical systems. Leveraging Dirac structures, instead of symplectic or Poisson structures, this formalism allows the incorporation of energy exchange within the spatial domain or through its boundary, which allows for a more comprehensive description of continuum mechanics. Building upon our recent work in describing nonlinear elasticity using exterior calculus and bundle-valued differential forms, this paper focuses on the systematic derivation of port-Hamiltonian models for solid and fluid mechanics in the material, spatial, and convective representations using Hamiltonian reduction theory. This paper also discusses constitutive relations for stress within this framework including hyper-elasticity, for both finite and infinitesimal strains, as well as viscous fluid flow governed by the Navier–Stokes equations.

We introduce a generalization of Möbius energy for knots to an energy functional for tubular neighbourhoods of closed inextensible curves. We prove the continuity of the energy and its boundedness for physically admissible tubes without self-contact and in particular for the torus. The functional allows to distinguish isotopy classes of the centreline through its physical resemblance to a self-repulsive electrostatic energy. If the tube has zero thickness, O’Hara’s functional is recovered. Finally, a discussion on the possible exponents in the functional is carried out and numerical examples are studied to highlight some advantages of our new definition.

We consider the Riemann–Hilbert correspondence associated with the q-difference sixth Painlevé equation in the crystal limit, i.e., q→0, and show two main results. First, the limit of this generically highly transcendental mapping is shown to exist. Second, we show that the limiting map is bi-rational and describe it explicitly.