American Institute of Mathematical Sciences

In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of \begin{document}$ \mathbb{R}^d $\end{document}, \begin{document}$ d \geqslant 3 $\end{document}. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process \begin{document}$ (\Phi, \mathcal{R}) $\end{document}. The point process \begin{document}$ \Phi $\end{document} generating the centres of the holes is either a Poisson point process or the lattice \begin{document}$ \mathbb{Z}^d $\end{document}; the marks \begin{document}$ \mathcal{R} $\end{document} generating the radii are unbounded i.i.d random variables having finite \begin{document}$ (d-2+\beta) $\end{document}-moment, for \begin{document}$ \beta > 0 $\end{document}. We study the rate of convergence to the homogenized solution in terms of the parameter \begin{document}$ \beta $\end{document}. We stress that, for low values of \begin{document}$ \beta $\end{document}, the balls generating the holes may overlap with overwhelming probability.

We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant \begin{document}$ -\alpha $\end{document}, with \begin{document}$ \alpha>0 $\end{document}. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behavior, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant \begin{document}$ \alpha $\end{document}, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.

We prove the convergence of the vanishing viscosity approximation for a class of \begin{document}$ 2\times2 $\end{document} systems of conservation laws, which includes a model of traffic flow in congested regimes. The structure of the system allows us to avoid the typical constraints on the total variation and the \begin{document}$ L^1 $\end{document} norm of the initial data. The key tool is the compensated compactness technique, introduced by Murat and Tartar, used here in the framework developed by Panov. The structure of the Riemann invariants is used to obtain the compactness estimates.

In this work, we study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit. The main difficulty is the presence of two parameters which requires us to obtain uniform bounds on the Bloch spectral data in various regimes of the parameter.

We study emergent behaviors of the Lohe Hermitian sphere(LHS) model with a time-delay for a homogeneous and heterogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit Hermitian sphere in \begin{document}$ \mathbb C^d $\end{document} with \begin{document}$ d \geq 2 $\end{document}. Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on \begin{document}$ \mathbb R^{2d} $\end{document}. In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.

A model of irrigation network, where lower branches must be thicker in order to support the weight of the higher ones, was recently introduced in [7]. This leads to a countable family of ODEs, describing the thickness of every branch, solved by backward induction. The present paper determines what kind of measures can be irrigated with a finite weighted cost. Indeed, the boundedness of the cost depends on the dimension of the support of the irrigated measure, and also on the asymptotic properties of the ODE which determines the thickness of branches.