Mathematical Models and Methods in Applied Sciences

In this work, we propose a method of opinion pooling based on pairwise interactions. We assume that each agent has a probability measure on the possible outcomes of some situation, and they try to find a single measure aggregating their estimates. This is a classical problem in Decision Theory, where expert opinions contain some degree of uncertainty, and a Decision Taker needs to pool these estimates. We study this problem using a kinetic theory approach, obtaining a Boltzmann type equation for opinions which are symmetric probability measures defined on the real line. We obtain a non-local, first order, mean field equation as its grazing limit when the parameter in the interaction goes to zero. Also, we prove the convergence to quasi-consensus with explicit estimates on the convergence time depending on the variance of these measures. Let us remark that this model can be interpreted as a noisy model of opinion dynamics. In many models, the opinion of each agent is a point in the real line, the agents interact and observe other agents opinions. We can consider that observed opinions are perturbed or deformed by some noise in the transmission channel or in the interpretation of the agents, so we can think of agents opinions directly as random variables instead of a single point.

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.

We consider a Landau–de Gennes model for a suspension of small colloidal inclusions in a nematic host. We impose suitable anchoring conditions at the boundary of the inclusions, and we work in the dilute regime — i.e. the size of the inclusions is much smaller than the typical separation distance between them, so that the total volume occupied by the inclusions is small. By studying the homogenised limit, and proving rigorous convergence results for local minimisers, we compute the effective free energy for the doped material. In particular, we show that not only the phase transition temperature, but also any coefficient of the quartic Landau–de Gennes bulk potential can be tuned, by suitably choosing the surface anchoring energy density.

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of two rarefaction waves and a viscous contact wave for the Cauchy problem to a one-dimensional compressible non-isentropic Navier–Stokes/Allen–Cahn system which is a combination of the classical Navier–Stokes system with an Allen–Cahn phase field description. We first construct the composite wave through Euler equations under the assumption of $\chi (x,t)\equiv 1$ for the large time behavior, and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier–Stokes/Allen–Cahn system. The proof is mainly based on a basic energy method.

We consider a hyperbolic–parabolic system arising from a chemotaxis model in tumor angiogenesis, which is described by a Keller–Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.