We introduce several new concepts called enhanced pullback attractors for nonautonomous dynamical systems by improving the compactness and attraction of the usual pullback attractors in strong topology spaces uniformly over some infinite time intervals. Then we establish several necessary and sufficient criteria for the existence, regularity and asymptotic stability of these enhanced pullback attractors

We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows for the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems

In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer’s fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further

This paper deals with an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy ut=Δu−∇⋅(u∇v)+μu(1−u)−uz,vt=−(u+w)v,wt=Δw−∇⋅(w∇v)−w+uz,zt=DzΔz−z−uz+βw, in a smoothly bounded domain Ω⊂ℝ3 with β>0, μ>0 and Dz>0. Based on a self-map argument, it is shown that under the assumption βmax{1,∥u0∥L∞(Ω)}<1+(1+1minx∈Ωu0(x))−1, this problem possesses

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 L2(Br(x)∩Ω)

In this paper, we consider the following system: nt+u⋅∇n=Δn−∇⋅(nχ(c)∇c),ct+u⋅∇c=Δc−cn,ut+κ(u⋅∇)u=Δu+∇P+n∇Φ, in a smoothly bounded domain Ω⊂ℝ2, with κ∈{0,1} and a given function χ(c)=1c𝜃 with 𝜃∈[0,1). It is proved that if κ=1 then for appropriately small initial data an associated no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in the classical sense, and that if κ=0

This preface describes motivational aspects related to a special issue focusing on “analysis of complex chemotaxis models”, and briefly discusses the contributions provided by the six papers contained therein.

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

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity ut=Δu−∇⋅(u(1+u)α−1∇v) and 0=Δv−⨍Ωudx+u, posed on Ω={x∈ℝn:|x|2n. Here we consider the supercritical case α≥2n and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass mc:=mc(n,R,α) such that (1) for arbitrary nonincreasing nonnegative initial data u0(x)=u0(|x|)

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

We consider the chemotaxis-Navier–Stokes system with gradient-dependent flux limitation and nonlinear production: nt+u⋅∇n=Δn−∇⋅(nf(|∇c|2)∇c), ct+u⋅∇c=Δc−c+g(n), ut+(u⋅∇)u+∇P=Δu+n∇ϕ and ∇⋅u=0 in a bounded domain Ω⊂ℝ2, where the flux limitation function f∈C2([0,∞]) and the signal production function g∈C1([0,∞]) generalize the prototypes f(s)=Kf(1+s)−α2 and g(s)=Kgs(1+s)β−1 with Kf,Kg>0, α∈ℝ and β>0.

A system of drift-diffusion equations for the electron, hole, and oxygen vacancy densities in a semiconductor, coupled to the Poisson equation for the electric potential, is analyzed in a bounded domain with mixed Dirichlet–Neumann boundary conditions. This system describes the dynamics of charge carriers in a memristor device. Memristors can be seen as nonlinear resistors with memory, mimicking the

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and

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

This paper deals with the micro–macro derivation of models from the underlying description provided by methods of the kinetic theory for active particles. We consider the so-called exotic models according to the definition proposed in [ N. Bellomo, N. Outada, J. Soler, Y. Tao and M. Winkler, Chemotaxis and cross diffusion models in complex environments: Modeling towards a multiscale vision, Math. Models

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

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

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

The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on p∈(1,2) (shear thinning case). For this problem we establish the existence of a weak solution

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

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

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

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

The stabilization parameters of the methods like the Streamline-Upwind/Petrov–Galerkin, Pressure-Stabilizing/Petrov–Galerkin, and the Variational Multiscale method typically involve two local length scales. They are the advection and diffusion length scales, appearing in the expressions for the advective and diffusive limits of the stabilization parameter. The advection length scale has always been

The main goal of this paper is to prove L1-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

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

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 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

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

In this paper, we develop stability analysis, including inverse inequality, L2 norm equivalence and interpolation error estimates, for divergence free conforming virtual elements in arbitrary dimension. A local energy projector based on the local Stokes problem is suggested, which commutes with the divergence operator. After defining a discrete bilinear form and a stabilization involving only the boundary

Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves (TWs) move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction–diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions

A multiscale model for glioma spread in brain tissue under the influence of vascularization and vascular endothelial growth factors is proposed. It accounts for the interplay between the different components of the neoplasm and the healthy tissue and it investigates and compares various therapy approaches. Precisely, these involve radiotherapy and chemotherapy in a concurrent or adjuvant manner together

We continue our study [R. Aylwin, C. Jerez-Hanckes, C. Schwab and J. Zech, Domain uncertainty quantification in computational electromagnetics, SIAM/ASA J. Uncertain. Quant.8 (2020) 301–341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping

Knowledge spillovers occur when a firm researches a new technology and that technology is adapted or adopted by another firm, resulting in a social value of the technology that is larger than the initially predicted private value. As a result, firms systematically under-invest in research compared with the socially optimal investment strategy. Understanding the level of under-investment, as well as

We study a regularized version of the Landau equation, which was recently introduced in [J. A. Carrillo, J. Hu, L. Wang and J. Wu, A particle method for the homogeneous Landau equation, J. Comput. Phys. X7 (2020) 100066, 24] to numerically approximate the Landau equation with good accuracy at reasonable computational cost. We develop the existence and uniqueness theory for weak solutions, and we reinforce

In this work, the concept of dual natural-norm for parametrized linear equations is used to derive residual-based a posteriori error bounds characterized by a 𝒪(1) stability constant. We translate these error bounds into very effective practical a posteriori error estimators for reduced basis approximations and show how they can be efficiently computed following an offline/ online strategy. We prove

We consider the motion of a viscous compressible and heat conducting fluid confined in the gap between two rotating cylinders (Taylor–Couette flow). The temperature of the cylinders is fixed but not necessarily constant. We show that the problem admits a time-periodic solution as soon as the ratio of the angular velocities of the two cylinders is a rational number.

We study fast and slow clustering dynamics of Cucker–Smale ensemble with internal phase dynamics via the Cucker–Smale–Kuramoto (in short, CSK) model. The CSK model describes the emergent dynamics of flocking particles with phase dynamics. It consists of the Cucker–Smale flocking model and the Kuramoto model, and their interplay is registered in the communication weight function between particles. We

We present a four-field virtual element discretization for the time-dependent resistive magnetohydrodynamics equations in three space dimensions, focusing on the semi-discrete formulation. The proposed method employs general polyhedral meshes and guarantees velocity and magnetic fields that are divergence free up to machine precision. We provide a full convergence analysis under suitable regularity

The strong asymptotic stabilization of 3D hyperbolic dynamics is achieved by a damped 2D elastic structure. The model is a Neumann wave-type equation with low regularity coupling conditions given in terms of a nonlinear von Karman plate. This problem is motivated by the elimination of aeroelastic instability (sustained oscillations of bridges, airfoils, etc.) in engineering applications. Empirical

This paper aims to develop numerical approximations of the Keller–Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a non-negative variable. We propose two algorithms, which combine a stabilized

In this paper, we study a singular limit problem for a compressible one-velocity bifluid system. More precisely, we show that solutions of the Kapila system generated by initial data close to equilibrium are obtained in the pressure-relaxation limit from solutions of the Baer–Nunziato (BN) system. The convergence rate of this process is a consequence of our stability result. Besides the fact that the

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant.

We investigate the uniform reshuffling model for money exchanges: two agents picked uniformly at random redistribute their dollars between them. This stochastic dynamics is of mean-field type and eventually leads to a exponential distribution of wealth. To better understand this dynamics, we investigate its limit as the number of agents goes to infinity. We prove rigorously the so-called propagation

A kinetic modeling of crowd evacuation with leaders and followers is considered in this paper, in which the followers may not know the full information about the walking environment and the evacuation strategy, but they follow the leaders and learn the walking strategy to get out of the walking venue. Based on the kinetic theory of active particles, the learning dynamics are considered by introducing

This paper deals with the modeling and numerical simulations of multilane vehicular traffic according to the kinetic theory of active particles methods. The main idea of this theory is to consider each driver–vehicle system as a micro-system, where the microscopic state of particles is described by position, velocity, and activity which is an appropriate variable for modeling the quality of the driver-vehicle

This editorial paper reviews the articles published in a special issue devoted to the application of active particle methods 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, crowd vehicle and crowd dynamics

Finite element methods for electromagnetic problems modeled by Maxwell-type equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to electromagnetic interface problems, as they are based

In this paper, we consider a coupled chemotaxis-fluid system that models self-organized collective behavior of oxytactic bacteria in a sessile drop. This model describes the biological chemotaxis phenomenon in the fluid environment and couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier–Stokes equations subject to a gravitational force

We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, lower semicontinuous energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain

In this paper, we consider the one-phase Stefan problem with surface tension, set in a two-dimensional strip-like geometry, with periodic boundary conditions respect to the horizontal direction x1∈𝕋. We prove that the system is locally null-controllable in any positive time, by means of a control supported within an arbitrary open and non-empty subset. We proceed by a linear test and duality, but

In this work, we prove the convergence of residual distribution (RD) schemes to dissipative weak solutions of the Euler equations. We need to guarantee that the RD schemes are fulfilling the underlying structure preserving methods properties such as positivity of density and internal energy. Consequently, the RD schemes lead to a consistent and stable approximation of the Euler equations. Our result

In this paper, we prove the uniqueness and structural stability of Poiseuille flows for axisymmetric solutions of steady Navier–Stokes system supplemented with Navier boundary conditions in a periodic pipe. Moreover, the stability is uniform with respect to both the flux and the slip coefficient of Navier boundary conditions. It is also shown that the nonzero frequency part of the velocity is bounded

Over the last decades, many diffuse-interface Navier–Stokes Cahn–Hilliard (NSCH) models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. The overarching objective of this work is to bring all of these models together by laying down a unified framework of NSCH models with non-zero mass fluxes

We introduce a new consensus-based optimization (CBO) method where an interacting particle system is driven by jump-diffusion stochastic differential equations (SDEs). We study well-posedness of the particle system as well as of its mean-field limit. The major contributions of this paper are proofs of convergence of the interacting particle system towards the mean-field limit and convergence of a discretized

We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence

Motivated by recent strain-limiting models for solids and biological fibers, we introduce the first intrinsic set of nonlinear constitutive relations, between the geometrically exact strains and the components of the contact force and contact couple, describing a uniform, hyperelastic, strain-limiting special Cosserat rod. After discussing some attractive features of the constitutive relations (orientation

In this paper, we present a rigorous derivation of the isothermal Euler-alignment model with singular communication weights. We consider a hydrodynamic limit of a kinetic Fokker–Planck-alignment model, which is the nonlinear Fokker–Planck equation with the Cucker–Smale alignment force. Our analysis is based on the estimate of relative entropy between macroscopic quantities, together with careful analysis

As a simplified version of a three-component taxis cascade model accounting for different migration strategies of two population groups in search of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional bounded convex domain with smooth boundary. For any given conveniently regular and biologically meaningful initial data, smallness conditions on the prescribed resource

We study the geometric ergodicity and the long-time behavior of the Random Batch Method for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB–IPS), the distribution laws converge to their respective invariant