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This editorial paper presents the papers published in a special issue devoted to the modeling and simulation of mutating virus pandemics in a globally connected world. The presentation is proposed in three parts. First, motivations and objectives are presented according to the idea that mathematical models should go beyond deterministic population dynamics by considering the multiscale, heterogeneous features of the complex system under consideration. Subsequently, the contents of the papers in this issue are presented referring to the aforementioned complexity features. Finally, a critical analysis of the overall contents of the issue is proposed, with the aim of providing a forward look to research perspectives. [ABSTRACT FROM AUTHOR]

We study the homogenization limit of the Mullins–Sekerka problem which serves as a model for late-stage coarsening in a phase transformation. We consider the case that the screening length which describes the effective range of particle interactions is much smaller than the system size, which leads to homogenization problems in unbounded domains. The present paper deals with deterministic initial particle distributions which are in an average sense homogeneously distributed. Stochastic particle distributions will be considered in a second paper (Part II). [ABSTRACT FROM AUTHOR]

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity u t = Δ u − ∇ ⋅ (u (1 + u) α − 1 ∇ v) and 0 = Δ v − ⨍ Ω u d x + u , posed on Ω = { x ∈ ℝ n : | x | < R } (n ≥ 2) and subjected to homogeneous Neumann boundary conditions. It is well-known that 2 n is the critical exponent of the systems in the sense that all solutions exist globally if α < 2 n and there exist finite-time blowup solutions if α > 2 n . Here we consider the supercritical case α ≥ 2 n and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass m c : = m c (n , R , α) such that (1) for arbitrary nonincreasing nonnegative initial data u 0 (x) = u 0 (| x |) with ∫ Ω u 0 > m c and u 0 ≢ ⨍ Ω u 0 , the corresponding solution blows up in finite time if α > 2 n , and if α = 2 n we can only prove that the solution blows up in finite time or infinite time; (2) for some nonincreasing nonnegative initial data with ∫ Ω u 0 < m c , the corresponding solutions are globally bounded. Our results extend that of Winkler's paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with α = 1. [ABSTRACT FROM AUTHOR]

This editorial paper is devoted to present the papers published in a special issue focused on modeling, qualitative analysis and simulation of the collective dynamics of living, self-propelled particles. A critical analysis of the overall contents of the issue is proposed, thus leading to a forward look to research perspectives. [ABSTRACT FROM AUTHOR]

This paper is concerned with the higher-dimensional haptotactic system modeling oncolytic virotherapy, which was initially proposed by Alzahrani–Eftimie–Trucu [Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci. 310 (2019) 76–95] (see also the survey Bellomo–Outada et al. [Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision, Math. Models Methods Appl. Sci. 32 (2022) 713–792]) to model the process of oncolytic viral therapy. We consider this problem in a bounded domain Ω ⊆ ℝ N (N = 2 , 3) with zero-flux boundary conditions. Although the L ∞ -norm of the extracellular matrix density v is easily obtainable, the remodeling process still causes difficulty due to the deficiency of regularity for v. Relying on some L p -estimate techniques, in this paper, under the mild condition on parameters, we finally established the existence of global-in-time classical solution, which is bounded uniformly. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when κ u = 0 , the corresponding solution of the system decays to (0 , 0 , 0) algebraically. To the best of our knowledge, these are the first results on boundedness and asymptotic behavior of the system in three-dimensional space. [ABSTRACT FROM AUTHOR]

We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch's framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble's framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions: (i) The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution. (ii) The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions. (iii) For the second-order fully implicit and convex splitting schemes, for any time step size δ t > 0 , there exists an initial condition u 0 , with | u 0 | > 1 , such that the numerical solution converges to the wrong steady state solution. (iv) For | u 0 | ≤ 1 , all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small. (v) An unconditionally energy-stable scheme (such as the modified Crank–Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank–Nicolson scheme). Most, if not all, of the above conclusions are expected to be true for more general Allen–Cahn and other phase-field models. [ABSTRACT FROM AUTHOR]

In materials that undergo martensitic phase transformation, distinct elastic phases often form layered microstructures — a phenomenon known as twinning. In some settings the volume fractions of the phases vary macroscopically; this has been seen, in particular, in experiments involving the bending of a bar. We study a two-dimensional (2D) model problem of this type, involving two geometrically nonlinear phases with a single rank-one connection. We adopt a variational perspective, focusing on the minimization of elastic plus surface energy. To get started, we show that twinning with variable volume fraction must occur when bending is imposed by a Dirichlet-type boundary condition. We then turn to paper's main goal, which is to determine how the minimum energy scales with respect to the surface energy density and the transformation strain. Our analysis combines ansatz-based upper bounds with ansatz-free lower bounds. For the upper bounds we consider two very different candidates for the microstructure: one that involves self-similar refinement of its length scale near the boundary, and another based on piecewise-linear approximation with a single length scale. Our lower bounds adapt methods previously introduced by Chan and Conti to address a problem involving twinning with constant volume fraction. The energy minimization problem considered in this paper is not intended to model twinning with variable volume fraction involving two martensite variants; rather, it provides a convenient starting point for the development of a mathematical toolkit for the study of twinning with variable volume fraction. [ABSTRACT FROM AUTHOR]

This paper focuses on Herbert A. Simon's visionary theory of the Artificial World. The artificial world evolves over time as a result of various actions, including interactions with the external world as well as interactions among its internal components. This paper proposes a mathematical theory of the conceptual framework of the artificial world. This goal requires the development of new mathematical tools, inspired in some way by statistical physics and stochastic game theory. The mathematical theory is applied in particular to the study of the dynamics of organizational learning, where cooperation and competition evolve through decomposition and recombination of organizational structures; the effectiveness of the evolutionary changes depends on the dynamic prevalence of cooperative over selfish behaviors, showing features common to the evolution of all living systems. [ABSTRACT FROM AUTHOR]

This editorial is dedicated to presenting the papers published in a special issue focused on modeling, qualitative analysis and simulation of the collective dynamics of systems in engineering and life sciences. All papers have a minor or major reference to living, i.e. complex systems, and a critical analysis of the overall content of the issue is proposed, leading to a forward look at research perspectives. This paper first defines the goals of the issue. Then, a brief description of the scientific contribution of the papers published in this issue is given. Finally, a look into the future is proposed. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of conforming virtual element methods (VEMs) for general second-order linear elliptic problems with rough source terms and applies it to a Poisson inverse source problem with rough measurements. For the forward problem, when the source term belongs to H − 1 (Ω) , the right-hand side for the discrete approximation defined through polynomial projections is not meaningful even for standard conforming VEM. The modified discrete scheme in this paper introduces a novel companion operator in the context of conforming VEM and allows data in H − 1 (Ω). This paper has three main contributions. The first contribution is the design of a conforming companion operator J from the conforming virtual element space to the Sobolev space V : = H 0 1 (Ω) , a modified virtual element scheme, and the a priori error estimate for the Poisson problem in the best-approximation form without data oscillations. The second contribution is the extension of the a priori analysis to general second-order elliptic problems with source term in V * . The third contribution is an application of the companion operator in a Poisson inverse source problem when the measurements belong to V * . Tikhonov's regularization technique regularizes the ill-posed inverse problem, and the conforming VEM approximates the regularized problem given a finite measurement data. The inverse problem is also discretized using the conforming VEM and error estimates are established. Numerical tests on different polygonal meshes for general second-order problems, and for a Poisson inverse source problem with finite measurement data verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers a system consisting of a number of interacting living entities whose state at the microscopic scale is heterogeneously distributed among the said entities. This state includes, in addition, the classical mechanical variables, such as position and velocity, also a behavioral variable which is modified by interactions. It is shown how the pioneering ideas proposed in Bellomo et al. [Towards a mathematical theory of behavioral swarms, ESAIM Control Optim. Calc. Var. 26 (2020) 125] can be developed towards modeling behavioral swarms within a quest towards a mathematical theory of living systems. The first part of the paper presents a qualitative analysis of the emerging behaviors predicted by the model in aforementioned work. Some simulations follow to depict the said emerging behaviors. The last part of the paper is devoted to derive a new, more general theory in view of applications to model living systems. [ABSTRACT FROM AUTHOR]

This paper first provides an introduction to the mathematical approach to the modeling, qualitative analysis, and simulation of large systems of living entities, specifically self-propelled particles. Subsequently, a presentation of the papers published in this special issue follows. Finally, a critical analysis of the overall contents of the issue is proposed, thus leading to define some challenging research perspectives. [ABSTRACT FROM AUTHOR]

We consider the chemotaxis-Navier–Stokes system with gradient-dependent flux limitation and nonlinear production: n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n f (| ∇ c | 2) ∇ c) , c t + u ⋅ ∇ c = Δ c − c + g (n) , u t + (u ⋅ ∇) u + ∇ P = Δ u + n ∇ ϕ and ∇ ⋅ u = 0 in a bounded domain Ω ⊂ ℝ 2 , where the flux limitation function f ∈ C 2 ([ 0 , ∞ ]) and the signal production function g ∈ C 1 ([ 0 , ∞ ]) generalize the prototypes f (s) = K f (1 + s) − α 2 and g (s) = K g s (1 + s) β − 1 with K f , K g > 0 , α ∈ ℝ and β > 0. For the linear production case of β = 1 , the global boundedness of solutions has been verified in the related literature for α > 0. In this paper, we expand to prove that the corresponding initial-boundary value problem possesses a unique globally bounded solution if α > (2 β − 1) + − 1 for 0 < β < 1 , or if α > 1 − 1 2 β − 1 for β > 1 , which shows that when 0 < β < 1 , that is, the self-enhancement ability of chemoattractant is weak, the solutions still remain globally bounded even though the flux limitation is relaxed to permit proper α ≤ 0 ; however, if β > 1 , it is necessary to impose the stronger flux limitation than that in the case β = 1 to inhibit the possible finite-time blow-up. This seems to be the first result on the global solvability in the chemotaxis-Navier–Stokes model with nonlinear production. [ABSTRACT FROM AUTHOR]

This paper deals with a parabolic–elliptic chemotaxis-consumption system with tensor-valued sensitivity S (x , n , c) under no-flux boundary conditions for n and Robin-type boundary conditions for c. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity S. One of the main steps is to show that ∇ c (⋅ , t) becomes tiny in L 2 (B r (x) ∩ Ω) for every x ∈ Ω ¯ and t when r is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity S = χ (x , n , c) , there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius R and all given data are radial. The result of the radial case covers scalar-valued sensitivity χ that can be singular at c = 0. [ABSTRACT FROM AUTHOR]

The Keller–Segel model is a paradigm to describe the chemotactic mechanism, which plays a vital role on the physiological and pathological activities of uni-cellular and multi-cellular organisms. One of the most interesting variants is the coupled system with the intrinsic growth, which admits many complex nontrivial patterns. This paper is devoted to the construction of multi-spiky solutions to the Keller–Segel models with the logistic source in 2D. Assuming that the chemo-attractive rate is large, we apply the inner-outer gluing scheme to nonlocal cross-diffusion system and prove the existence of multiple boundary and interior spikes. The numerical simulations are presented to highlight our theoretical results. [ABSTRACT FROM AUTHOR]

Two fundamental models in plasma physics are given by the Vlasov–Maxwell–Boltzmann system and the compressible Euler–Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number tends to zero. In this paper, we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new -dependent energy estimates basing on the macro–micro decomposition to characterize the asymptotic limit in the compressible setting. [ABSTRACT FROM AUTHOR]

In recent years, isogeometric analysis (IGA) has attracted significant attention from the computational mechanics community due to its ability to integrate design and analysis. Besides, IGA is also a higher-order discretization technique for solving partial differential equations, showing high approximation capability per degree of freedom. In this paper, we extend the application realm of IGA to particle-laden flows based on Eulerian–Eulerian description that couples Navier–Stokes equations with a density transport equation through a Boussinesq approximation. The coupled systems are solved by using quadratic non-uniform rational B-spline (NURBS) functions and a recently developed residual-based variational multiscale (VMS) formulation, which introduces coupling between the fine velocity scales and density equation residuals. We deploy the proposed approach to perform large-eddy simulations (LES) of dilute particle-laden flows over a flat surface at Reynolds number = 10,000. We compare the simulation results against direct numerical simulation (DNS) results from the literature. We find that combining VMS and IGA, the proposed approach enables accurate prediction of a wide range of flow/particle statistics with a relatively lower mesh resolution. [ABSTRACT FROM AUTHOR]

This paper deals with the micro–macro-derivation of virus models coupled with a reaction–diffusion system that generates the dynamics in space of the virus particles. The first part of the presentation focuses, starting from [N. Bellomo, K. Painter, Y. Tao and M. Winkler, Occurrence versus absence of taxis-driven instabilities in a May–Nowak model for virus infection, SIAM J. Appl. Math. 79 (2019) 1990–2010; N. Bellomo and Y. Tao, Stabilization in a chemotaxis model for virus infection, Discrete Contin. Dyn. Syst. S 13 (2020) 105–117], on a survey and a critical analysis of some phenomenological models known in the literature. The second part shows how a Hilbert type can be developed to derive models at the macro-scale from the underlying description delivered by the kinetic theory of active particles. The third part deals with the derivation of macroscopic models of various virus models coupled with the reaction–diffusion systems. Then, a forward look to research perspectives is proposed. [ABSTRACT FROM AUTHOR]

The contents of this editorial is proposed in three parts devoted to the modeling and simulation of mutating virus pandemics in a globally connected world. The presentation is as follows: first, a general framework is presented according to the idea that mathematical models should go beyond deterministic population dynamics by considering the multiscale, heterogeneous features of the complex system under consideration; second, the contents of the papers in this issue are presented; finally, there is a critical analysis looking ahead to research perspectives. [ABSTRACT FROM AUTHOR]

In this paper, we study a gang territorial model consisting of two parabolic and two ordinary differential equations, where a taxis-type mechanism models that the two rivaling gangs are repelled by each other's graffiti. Our main analytical finding shows the existence of global, bounded classical solutions. By making use of quantitative global estimates, we prove that these solutions converge to homogeneous steady states if the initial data are sufficiently small. Moreover, we perform numerical experiments which show that for different choices of parameters, the system may become diffusion- or convection-dominated, where in the former case the solutions converge toward constant steady states while in the latter case nontrivial asymptotic behavior such as segregation is observed. In order to perform these experiments, we apply a nonlinear finite element flux-corrected transport method (FEM-FCT) which is positivity-preserving. Then we treat the nonlinearities in both the system and the proposed nonlinear scheme simultaneously using fixed-point iteration. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following Keller–Segel system with flux limitation u t = ∇ ⋅ u ∇ u u 2 + | ∇ u | 2 − χ ∇ ⋅ (u f (| ∇ v | 2) ∇ v) , x ∈ Ω ,   t > 0 , 0 = Δ v − μ + u , x ∈ Ω ,   t > 0 , (⋆) under no-flux boundary conditions in a ball Ω = B R (0) ⊂ ℝ n (n ≥ 1), where μ : = 1 | Ω | ∫ Ω u 0 (x) d x is positive and f (ξ) = (1 + ξ) − α , for ξ ≥ 0 with α > 0. It is proved that the problem (⋆) possesses a unique classical solution that can be extended in time up to a maximal T max ∈ (0 , ∞ ]. Moreover, it is shown that the above solution is global and bounded when either χ < (1 + m 2) α m if n = 1 and α ≤ 1 2 , or χ < 2 α − 1 (2 α 2 α − 1) α if n ≥ 1 and α > 1 2 . We point out that when α = 1 2 , our result is consistent with that of [N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436–473]. [ABSTRACT FROM AUTHOR]

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: u t = u x x − χ u v v x x + u (a − b u) , 0 < x < L , t > 0 , 0 = v x x − μ v + ν u , 0 < x < L , t > 0 , u x (t , 0) = u x (t , L) = v x (t , 0) = v x (t , L) = 0 t > 0 , where χ , a , b , μ , ν are positive constants. Among others, we prove there are χ ∗ > 0 and { χ k ∗ } ⊂ [ χ ∗ , ∞) ( χ ∗ ∈ { χ k ∗ }) such that the constant solution (a b , ν μ a b) of system is locally stable when 0 < χ < χ ∗ and is unstable when χ > χ ∗ , and under some generic condition, for each k ≥ 1 , a (local) branch of nonconstant stationary solutions of system bifurcates from (a b , ν μ a b) when χ passes through χ k ∗ , and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions { (u (⋅ ; χ n) , v (⋅ ; χ n)) } of system with χ = χ n (→ ∞) develops spikes at any x ∗ satisfying liminf n → ∞ u (x ∗ ; χ n) > a b . Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from (a b , ν μ a b) when χ passes through χ ∗ can be extended to χ = ∞ and the stationary solutions on this global bifurcation extension are locally stable when χ ≫ 1 and develop spikes as χ → ∞. [ABSTRACT FROM AUTHOR]

This paper shows how a new theory of epidemics can be developed for viral pandemics in a globally interconnected world. The study of the in-host dynamics and, in parallel, the spatial diffusion of epidemics defines the goal of our work, which looks ahead to new mathematical tools to model epidemics beyond the traditional approach of population dynamics. The approach takes into account the evolutionary nature of the virus, which can generate pseudo-Darwinian mutations and selection, while learning the presence of the virus and activating adaptive immunity. The study of immune competition plays a key role in both the in-host dynamics and the contagion dynamics. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of O (h k) in the energy norm, where k represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach. [ABSTRACT FROM AUTHOR]

In this paper, 3D–2D-dimensional reduction for hyperelastic thin films modeled through energies with point-dependent growth, assuming that the sample is clamped on the lateral boundary, is performed in the framework of Γ -convergence. Integral representation results, with a more regular Lagrangian related to the original energy density, are provided for the lower dimensional limiting energy, in different contexts. [ABSTRACT FROM AUTHOR]

In this paper, we present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic theory model describes, at the mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and the motion of immune cells, stimulated by cytokines. Moreover, we reproduce the consumption of myelin sheath due to anomalously activated lymphocytes and its restoration by oligodendrocytes. Successively, we fix a small time parameter and assume that the considered processes occur at different scales. This allows us to perform a formal limit, obtaining macroscopic reaction–diffusion equations for the number densities with a chemotaxis term. A natural step is then to study the system, inquiring about the formation of spatial patterns through a Turing instability analysis of the problem and basing the discussion on the microscopic parameters of the model. In particular, we get spatial patterns oscillating in time that may reproduce brain lesions characteristic of different phases of the pathology. [ABSTRACT FROM AUTHOR]

Celyvir is an advanced therapy medicine, consisting of mesenchymal stem cells (MSCs) containing the oncolytic virus ICOVIR 5. This paper sets out a dynamic system which attempts to capture the fundamental relationships between cancer, the immune system and adenoviruses. Two forms of treatment were studied: continuous and periodic, the second being closer to the real situation. In the analysis of the first model, in addition to identifying the critical points, their properties and bifurcation points, a number of numerical simulations were carried out. It has thus been shown that there are bistability regimes in which Celyvir can produce an equilibrium of tumor progression, or even freedom from tumor. A sensitivity analysis was also performed to determine which parameters are most important in the system. Subsequently, an optimal control problem with nonlinear objective functional has been formulated, where the therapeutic goal is not only to minimize the size of the tumor cell population and the total cost of treatment, but also to prevent the tumor from reaching a critical size. It has been shown that the optimal control is bang–bang. With the second model, a threshold value of viral load has been identified at which the success of the treatment could be ensured. It is clear in both models that a low viral load would lead to relapse of the disease. Finally, it is shown that a periodic bang–bang regime should be used to optimize treatment with Celyvir. [ABSTRACT FROM AUTHOR]

Social behavior in crowds, such as herding or increased interpersonal spacing, is driven by the psychological states of pedestrians. Current macroscopic crowd models assume that these are static, limiting the ability of models to capture the complex interplay between evolving psychology and collective crowd dynamics that defines a "social crowd". This paper introduces a novel approach by explicitly incorporating an "activity" variable into the modeling framework, which represents the evolving psychological states of pedestrians and is linked to crowd dynamics. To demonstrate the role of activity, we model pedestrian egress when this variable captures stress and awareness of contagion. In addition, to highlight the importance of dynamic changes in activity, we examine a scenario in which an unexpected incident necessitates alternative exits. These case studies demonstrate that activity plays a pivotal role in shaping crowd behavior. The proposed modeling approach thus opens avenues for more realistic macroscopic crowd descriptions with practical implications for crowd management. [ABSTRACT FROM AUTHOR]

In this paper, we study a discrete momentum consensus-based optimization (Momentum-CBO) algorithm which corresponds to a second-order generalization of the discrete first-order CBO [S.-Y. Ha, S. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci. 30 (2020) 2417–2444]. The proposed algorithm can be understood as the modification of ADAM-CBO, replacing the normalization term by unity. For the proposed Momentum-CBO, we provide a sufficient framework which guarantees the convergence of algorithm toward a global minimum of the objective function. Moreover, we present several experimental results showing that Momentum-CBO has an improved success rate of finding the global minimum compared to vanilla-CBO and show the stability of Momentum-CBO under different initialization schemes. We also show that Momentum-CBO can be used as the alternative of ADAM-CBO which does not have a proper convergence analysis. Finally, we give an application of Momentum-CBO for Lyapunov function approximation using symbolic regression techniques. [ABSTRACT FROM AUTHOR]

In this paper we study Nash equilibria in auctions from the kinetic theory of active particles point of view. We propose a simple learning rule for agents to update their bidding strategies based on their previous successes and failures, in first-price auctions with two bidders. Then, we formally derive the corresponding kinetic equations which describe the evolution over time of the distribution of agents on the bidding strategies. We show that the stationary solution of the equation corresponds to the symmetric Nash equilibrium of the auction, and we prove the convergence to this stationary solution when time goes to infinity. We also introduce a more general learning rule that only depends on the income of agents, and we apply to both first- and second-price auctions. We show that agents learn the Nash equilibrium in first- and second-price auctions with these rules. We present agent-based simulations of the models, and we discuss several open problems. [ABSTRACT FROM AUTHOR]

This editorial paper reviews the articles published in a special issue devoted to the application of active particle methods applied to the study of the collective dynamics of large systems of interacting entities in science and society. The applications presented in this special issue focus on the study of financial markets, cell dynamics in the context of cancer modeling, vehicle and crowd vehicle and crowd dynamics, and classical problems in the kinetic theory of active particles and swarm theory. A critical analysis is proposed to look forward to research. Perspectives with emphasis on the interaction of multiple social dynamics. [ABSTRACT FROM AUTHOR]

Geymonat extended Gurtin's result on Beltrami's completeness to L 2 -case. The first objective of this paper is to generalize Geymonat's result to L p -case with tangential and normal boundary conditions. Maggiani et al. proposed an extension of Beltrami's-type decomposition for symmetric matrix fields in s p (Ω) when Ω is simply connected domain and of class ∞ . The second objective of this paper is to present more extensions of the above decomposition when the domain Ω is only of class 1 , 1 and not necessarily simply connected. The third objective of this paper is to use the previous characterizations of matrix fields to solve some variants of the bi-Laplacian problem. [ABSTRACT FROM AUTHOR]

Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper, we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of C 1 isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We replace the hypothesis of local linear independence for the basis of each level by a weaker assumption, which still ensures the linear independence of hierarchical splines. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by C 1 splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems. [ABSTRACT FROM AUTHOR]

This paper presents a survey and critical analysis of the mathematical literature on modeling and simulation of human crowds taking into account behavioral dynamics. The main focus is on research papers published after the review [N. Bellomo and C. Dogbè, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev. 53 (2011) 409–463], thus providing important research perspectives related to new, emerging trends. The presentation addresses the scaling problem corresponding to microscopic (individual-based), mesoscopic (kinetic), and macroscopic (hydrodynamic) modeling and analysis. A multiscale vision guides the overall content of the paper. The critical analysis of the overall content naturally leads to research perspectives. A selection of them is brought to the attention of the interested reader together with hints on how to deal with them. [ABSTRACT FROM AUTHOR]

The aim of this paper is to elucidate the existence of patterns for Keller–Segel-type models that are solutions of the traveling pulse form. The idea is to search for transport mechanisms that describe this type of waves with compact support, which we find in the so-called nonlinear diffusion through saturated flux mechanisms for the movement cell. At the same time, we analyze various transport operators for the chemoattractant. The techniques used combine the analysis of the phase diagram in dynamic systems together with its counterpart in the system of partial differential equations through the concept of entropic solution and the admissible jump conditions of the Rankine–Hugoniot type. We found traveling pulse waves of two types that correspond to those found experimentally. [ABSTRACT FROM AUTHOR]

We introduce and analyze a discontinuous Galerkin method for the numerical modeling of the equations of Multiple-Network Poroelastic Theory (MPET) in the dynamic formulation. The MPET model can comprehensively describe functional changes in the brain considering multiple scales of fluids. Concerning the spatial discretization, we employ a high-order discontinuous Galerkin method on polygonal and polyhedral grids and we derive stability and a priori error estimates. The temporal discretization is based on a coupling between a Newmark β -method for the momentum equation and a -method for the pressure equations. After the presentation of some verification numerical tests, we perform a convergence analysis using an agglomerated mesh of a geometry of a brain slice. Finally, we present a simulation in a three-dimensional patient-specific brain reconstructed from magnetic resonance images. The model presented in this paper can be regarded as a preliminary attempt to model the perfusion in the brain. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to prove L 1 -comparison and contraction principles for weak solutions of PDE system corresponding to a phase transition diffusion model of Hele-Shaw type with addition of a linear drift. The flow is considered with a source term and subject to mixed homogeneous boundary conditions: Dirichlet and Neumann. The PDE can be focused to model for instance biological applications including multi-species diffusion-aggregation models and pedestrian dynamics with congestion. Our approach combines DiPerna-Lions renormalization type with Kruzhkov device of doubling and de-doubling variables. The L 1 -contraction principle allows afterwards to handle the problem in a general framework of nonlinear semigroup theory in L 1 , thus taking advantage of this strong theory to study furthermore existence, uniqueness, comparison of weak solutions, L 1 -stability as well as many further questions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized inverse iteration for computing ground states of the Gross–Pitaevskii eigenvector (GPE) problem. For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross–Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput.25 (2004) 1674–1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal.58 (2020) 1744–1772]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation. As a consequence, the plate shows non-flat deformations in equilibrium with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the liquid crystal orientation. We focus on thin plates in the bending regime and derive a two-dimensional bending model that combines a nonlinear bending energy for the deformation, with a surface Oseen–Frank energy for the director field that describes the local orientation of the liquid crystal elastomer. Both energies are nonlinearly coupled by means of a spontaneous curvature term that effectively describes the nematic-elastic coupling. We rigorously derive this model as a Γ -limit from three-dimensional, nonlinear elasticity. We also devise a new numerical algorithm to compute stationary points of the two-dimensional model. We conduct numerical experiments and present simulation results that illustrate the practical properties of the proposed scheme as well as the rich mechanical behavior of the system. [ABSTRACT FROM AUTHOR]

In this paper, we present a survey of recent progress on the emergent behaviors of stochastic particle models which arise from the modeling of collective dynamics. Collective dynamics of interacting autonomous agents is ubiquitous in nature, and it can be understood as a formation of concentration in a state space. The jargons such as aggregation, herding, flocking and synchronization describe such concentration phenomena. Recently it became one of the emerging topics in the applied mathematics community due to possible engineering applications and close relation with nonlocal partial differential equations. When an autonomous agent system interacts with unknown environment as an open system, the effects of hidden and unidentified interactions between the environment and the autonomous system are often realized by stochastic noises in agent dynamics, and the temporal evolution of the autonomous system results in stochastic collective models. From the viewpoint of dynamical systems theory, it is very interesting how collective dynamics emerges from initial state. As concrete examples, we consider four specific stochastic collective models (stochastic Winfree and Kuramoto models for synchronization, the stochastic Cucker–Smale model for flocking, and a first-order stochastic nonlinear consensus model), and we also briefly review the state-of-the-art results for these models on the emergence of collective dynamics and discuss their applications in finance and optimization. [ABSTRACT FROM AUTHOR]

This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a non-conforming spatial approximation and finite differences in time (Rothe–Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Therefore, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We formulate the problem in the very general and abstract setting of so-called non-conforming Bochner pseudo-monotone operators, which allows for a unified treatment of several evolution problems. Our abstract results for non-conforming Bochner pseudo-monotone operators allow to establish (weak) convergence just by verifying a few natural assumptions on the operators time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be performed easily. We exemplify the applicability of our approach on several DG schemes for the unsteady p -Navier–Stokes problem. The results of some numerical experiments are reported in the final section. [ABSTRACT FROM AUTHOR]

Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D'Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci107 (2008) 1249–1267; M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst.9 (2010) 462–483], Short et al. proposed an agent-based statistical model of residential burglary to model the crime hotspot phenomena. From the point of view of reaction–diffusion systems, the model is a chemotactic system with cross diffusion that exhibit hotspot phenomena. In this paper, we first construct a radial hotspot solution of this system, then study the linear stability of this hotspot solution by studying a nonlocal eigenvalue problem. It turns out that the stability of the hotspot is completely different depending on which spatial dimension the system is on. The main mathematical difficulty of the system involves treating the steep change of diffusion near the core of the hotspot, because of the quasilinearity induced by the cross diffusion. We believe that the techniques used in this paper can be developed to treat many other chemotactic systems. [ABSTRACT FROM AUTHOR]

This paper is a brief introduction to the mathematical tools employed in the modeling and simulation of the interactions of a large number of living entities, and provides a presentation of the papers published in a special issue devoted to this topic. In addition, we bring to the reader’s attention some perspective ideas on possible objectives of future researches. [ABSTRACT FROM AUTHOR]

The first part of our paper presents a general survey on the modeling, analytic problems, and applications of the dynamics of human crowds, where the specific features of living systems are taken into account in the modeling approach. This critical analysis leads to the second part which is devoted to research perspectives on modeling, analytic problems, multiscale topics which are followed by hints towards possible achievements. Perspectives include the modeling of social dynamics, multiscale problems and a detailed study of the link between crowds and swarms modeling. [ABSTRACT FROM AUTHOR]

This editorial paper is devoted to present the papers published in a special issue focused on recent views, applications, and results of virtual element methods (VEM). A critical analysis of the overall contents of the issue is proposed, thus leading to a forward look to research perspectives. [ABSTRACT FROM AUTHOR]

This paper considers the discretization of the Stokes equations with Scott–Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf–sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order k of the Scott–Vogelius velocity space with appropriately chosen and explicitly given Raviart–Thomas bubbles. This approach is inspired by [X. Li and H. Rui, A low-order divergence-free H (div) -conforming finite element method for Stokes flows, IMA J. Numer. Anal.42 (2022) 3711–3734], where the case k = 1 was studied. The proposed method is pressure-robust, with optimally converging H 1 -conforming velocity and a small H (div) -conforming correction rendering the full velocity divergence-free. For k ≥ d , with d being the dimension, the method is parameter-free. Furthermore, it is shown that the additional degrees of freedom for the Raviart–Thomas enrichment and also all non-constant pressure degrees of freedom can be eliminated, effectively leading to a pressure-robust, inf–sup stable, optimally convergent P k × P 0 scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under s -Gevrey assumptions on the residual equation, we establish s -Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input. [ABSTRACT FROM AUTHOR]