Mathematical Models and Methods in Applied Sciences

Social behaviors play a crucial role in shaping wealth distribution in societies, while wealth distribution also significantly inuences social behaviors. To model the co-evolutionary multi-dynamics of wealth distribution and social behaviors, this study presents a kinetic model of wealth distribution with adaptive social behaviors. The model is developed within a semi-discrete framework of kineti…

This paper is concerned with a system that can be used to model the diffusion-advection of myosin molecules, which change direction by a certain angle when binding to actin gel. More precisely, we shall consider the parabolic-elliptic Keller-Segel system with rotation [Formula: see text]; [Formula: see text] in smoothly bounded planar domains, where [Formula: see text] is a matrix attaining value…

This paper investigates the large-time behavior of the viscous shock profile for the one-dimensional system of viscoelasticity, subject to initial perturbations that approach space-periodic functions at far fields. We specifically address the case with non-convex constitutive stress relations and non-degenerate Lax’s shock. Under the assumptions of suitably small initial perturbations satisfying …

The principle of conservation of energy implies that a crack growing in a viscoelastic body must satisfy the dynamic energy-dissipation balance, an equality involving the energy dissipated by viscosity and by crack growth. Unfortunately, some models of evolution of a viscoelastic body, like the frequently used Kelvin-Voigt model, imply that the dynamic energy-dissipation balance prevents crack gr…

In this paper, we consider the problem of energy conservation for weak solutions of the inviscid Primitive Equations (PE) in a bounded domain. Based on the work [Bardos et al., Onsager–s conjecture with physical boundaries and an application to the vanishing viscosity limit, Comm. Math. Phys., 2019, 291-310], we prove the energy conservation for PE with boundary condition under suitable Onsager-t…

Following the approach of Eckhaus, Mielke, and Schneider for reaction diffusion systems, we justify rigorously the Eckhaus stability criterion for stability of convective Turing patterns, as derived formally by complex Ginzburg-Landau approximation. Notably, our analysis includes higher-order, nonlocal, and even certain semilinear hyperbolic systems.

The paper is concerned with the mathematical analysis of a class of thermodynamically consistent kinetic models for nonisothermal flows of dilute polymeric fluids, based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid under consideration. The model involves a system of nonlinear partial differential equations coupling the unsteady incompressible t…

We consider fluid flow across a permeable interface within a deformable porous medium. We use mixture theory. The mixture’s constituents are assumed to be incompressible in their pure form. We use Hamilton’s principle to obtain the governing equations, and we propose a corresponding finite element implementation. The filtration velocity and the pore pressure are allowed to be discontinuous across…

We explore recent progress and open questions concerning local minima and saddle points of the Cahn–Hilliard energy in [Formula: see text] and the critical parameter regime of large system size and mean value close to [Formula: see text]. We employ the String Method of E, Ren, and Vanden-Eijnden—a numerical algorithm for computing transition pathways in complex systems—in [Formula: see text] to g…

This editorial is dedicated to recent developments and perspectives of mathematical tools for the modeling and applications for collective dynamics of active particles. These scientific contributions and the critical analysis proposed in this paper aim to propose a forward look to perspectives concerning both further developments of the mathematical theory and exploring new conceivable applicatio…

This survey collects, within a unified framework, various results (primarily by the authors themselves) on the use of Deterministic Infinite-Dimensional Optimal Control Theory to address applied economic models. The main aim is to illustrate, through several examples, the typical features of such models (including state constraints, non-Lipschitz data, and non-regularizing differential operators)…

In this paper, we propose a mathematical model for tumor invasion supported by angiogenesis and interactions with the surrounding tissue. For the model deduction we employ a multiscale approach starting from lower scales and obtaining by an informal parabolic upscaling a system of reaction–diffusion-taxis equations with a so-called ’taxis cascade’, where one species is performing taxis toward a s…

This paper deals with the following chemotaxis system with gradient-dependent flux limitation and nonlinear diffusion [Formula: see text] under homogeneous Neumann boundary conditions in a smoothly bounded domain [Formula: see text] [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] generalizes the prototype given by [Formula: see text] for all [Formula: see …

We consider a partial differential equation (PDE) model to predict residential burglary derived from a probabilistic agent-based model through a mean-field limit operation. The PDE model is a nonlinear, coupled system of two equations in two variables (attractiveness of residential sites and density of criminals), similar to the Keller–Segel model for aggregation based on chemotaxis. Unlike previ…

The differential model proposed in this paper is designed to capture the complex dynamics of viral infections and immune responses. Its novelty lies in the inclusion of messenger sub-particles that integrate interactions at molecular, cellular, and tissue levels. This paper’s innovative approach is to use kinetic theory in a multiscale framework. We examine the role of molecular messengers in sha…

Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback–Leibler (KL) divergence, originally introduced in kinetic theory as a measure of relative entropy between probability distributions. Just as in machine learning, the ability to quantify the proximity of probability distrib…

Mathematical modeling of virus dynamics is key to depicting the evolutionary pathways that lead to virus emergence, transmission, and persistence. Typically, viruses are populations of closely related genomes that continuously change their configuration and adapt to environmental selection pressures. In this work, we revisit this idea by considering viruses as active particles that dynamically sh…

We study the consensus dynamics of discrete-time behavioral swarm (DBS) models with random batch interactions and external noises. The proposed models describe behavioral swarms where each particle is characterized by its activity (level), spatial position, and heading angle. Interactions among particles are governed by the random batch method (RBM), which significantly reduces computational comp…

This paper provides a critical overview of the mathematical kinetic theory of active particles, which is used to model and study collective systems consisting of interacting living entities, such as those involved in behavior and evolution. The main objective is to study the interactions of large systems of living entities mathematically. More specifically, the study relates to the complex featur…