Mathematical Models and Methods in Applied Sciences

We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the Cucker–Smale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.

The paper introduces a new global–local method for recovering the microscale features of gradient fields inside heterogeneous media. The approach is based upon the L²-projection of the homogenized solution onto function spaces spanned by solutions of local problems. The projection framework introduced is general and applies to local domains ω on the interior of the domain of interest as well as those touching the boundary. Within this new framework it is shown that the difference between the actual solution and the L²-projection of the homogenized solution as measured in the energy norm over ω is the same order as the error between the homogenized solution and the actual solution as measured by the L²-norm over a slightly larger domain. In light of the classic results from periodic homogenization this is the best one can do see A. Besounssan, J. L. Lions and G. C. Papanicolau, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).

The Brinkman equations describe the flow of a viscous fluid in a porous matrix. Mathematically the Brinkman model is a parameter-dependent combination of both the Darcy and Stokes models. We introduce a dual mixed framework for the problem, and use H(div)-conforming finite elements with the symmetric interior penalty Galerkin method to obtain a stable formulation. We show that the formulation is stable in a mesh-dependent norm for all values of the parameter. We also introduce a postprocessing scheme for the pressure along with a residual-based a posteriori estimator, which is shown to be efficient and reliable for all parameter values.

This paper is devoted to hydrodynamic limits for collisional linear kinetic equations. It is a classical result that under certain conditions on the collision operator, the long time/small mean free path asymptotic behavior of the density of particles can be described by diffusion-type equations. We are interested in situations in which the degeneracy of the collision frequency for small velocities causes this limit to break down. We show that the appropriate asymptotic analysis leads to an anomalous diffusion regime.

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a Kac–Zwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M^-1/2) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small friction and o(M^-1/2) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuation–dissipation relation.

We study nonlinear waves in Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schrödinger (DpS) equation. It consists of a spatial discretization of a generalized Schrödinger equation with p-Laplacian, with fractional p > 2 depending on the exponent of Hertz's contact force. We show that the DpS equation admits explicit periodic traveling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov–Schmidt technique, we prove the existence of exact periodic traveling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime, namely modulational instabilities, the existence of static and traveling breathers, and repulsive or attractive interactions of these localized structures.