Authors:L. Pedraza, J. P. Pinasco, N. Saintier Pages: 1 - 36 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. 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. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-02-11T08:00:00Z DOI: 10.1142/S0218202520500062
Authors:Gregor Gantner, Dirk Praetorius, Stefan Schimanko Pages: 1 - 47 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. 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. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-02-05T08:00:00Z DOI: 10.1142/S0218202520500074
Authors:Kyudong Choi, Moon-Jin Kang, Young-Sam Kwon, Alexis F. Vasseur Pages: 1 - 51 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. 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 [math]-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. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-01-28T08:00:00Z DOI: 10.1142/S0218202520500104
Authors:Ting Luo, Haiyan Yin, Changjiang Zhu Pages: 1 - 43 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. 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 [math] 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. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-12-03T08:16:49Z DOI: 10.1142/S0218202520500098
Authors:Giacomo Canevari, Arghir Zarnescu Pages: 1 - 34 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. 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. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-11-25T02:41:32Z DOI: 10.1142/S0218202520500086
Authors:Bouchra Aylaj, Nicola Bellomo, Livio Gibelli, Alessandro Reali Pages: 1 - 22 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This paper proposes a multiscale vision to human crowds which provides a consistent description at the three possible modeling scales, namely, microscopic, mesoscopic, and macroscopic. The proposed approach moves from interactions at the microscopic scale and shows how the same modeling principles lead to kinetic and hydrodynamic models. Hence, a unified framework is developed which permits to derive models at each scale using the same principles and similar parameters. This approach can be used to simulate crowd dynamics in complex environments composed of interconnected areas, where the most appropriate scale of description can be selected for each area. This offers a pathway to the development of a multiscale computational model which has the capability to optimize the granularity of the description depending on the pedestrian local flow conditions. An important feature of the modeling at each scale is that the complex interaction between emotional states of walkers and their motion is taken into account. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-11-14T09:17:16Z DOI: 10.1142/S0218202520500013
Authors:Irene M. Gamba, Leslie M. Smith, Minh-Binh Tran Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-11-14T09:17:15Z DOI: 10.1142/S0218202520500037
Authors:Lukas Herrmann, Kristin Kirchner, Christoph Schwab Pages: 1 - 43 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [math] or, more generally, by a compact metric space [math] such as a compact [math]-manifold [math]. A colored GRF [math], admissible for our algorithms, solves the stochastic fractional-order equation [math] for some [math], where [math] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [math] is white noise on [math]. We thus consider GRFs on [math] with covariance operators of the form [math]. The proposed algorithms numerically approximate samples of [math] on nested sequences [math] of regular, simplicial partitions [math] of [math] and [math], respectively. Work and memory to compute one approximate realization of the GRF [math] on the triangulation [math] of [math] with consistency [math], for some consistency order [math], scale essentially linearly in [math], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [math] to white noise [math]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-11-14T09:17:15Z DOI: 10.1142/S0218202520500050
Authors:Shu Wang, Teng Wang Pages: 1 - 82 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We investigate the time-asymptotic stability of planar rarefaction wave for the 3D bipolar Vlasov–Poisson Boltzmann (VPB) system, based on the micro–macro decompositions introduced in [T. P. Liu and S. H. Yu, Boltzmann equation: Micro–macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004) 133–179; Energy method for the Boltzmann equation, Physica D 188 (2004) 178–192] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of basic wave patterns for bipolar VPB system in three dimensions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-11-14T09:17:14Z DOI: 10.1142/S0218202520500025
Authors:Wenjun Wang, Huanyao Wen Pages: 1 - 41 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. S[math]li, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci. 15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to [math], in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing [math]-norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-11-14T09:17:13Z DOI: 10.1142/S0218202520500049
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