In the space-time (ST) computational analysis, the discretization methods, such as those with finite elements, have appeared in a number of variations over the past years. Most of the current methods employ discontinuous functions in time. The motivation for that is to avoid a cost associated with fully 4D computations. Consequently, the increase in the number of unknowns in time remains under control despite using ST elements. Such methods have been applied over the past decades to a very large number of fluid dynamics and advection-diffusion problems, which are governed by equations with first-order time derivatives, and to a far lesser extent to elastodynamics and other solid mechanics problems, which are governed by equations with second-order time derivatives. In both contexts, especially when the functions are continuous in space, the ST methods are often formulated in the framework of stabilized methods, such as the Streamline-Upwind/Petrov-Galerkin, Galerkin/Least-Squares, Pressure-Stabilizing/Petrov-Galerkin, and variational multiscale methods. These are sometimes supplemented with discontinuity-capturing techniques. With the widespread adoption of isogeometric analysis (IGA), the ST methods have also been synthesized with IGA and applied to a large number of problems, mostly in fluid dynamics. The synthesis, ST-IGA, enables and encourages the use of higher-order functions in time. Within the ST domain, increasing the polynomial order in time leads to the expected improvement in solution accuracy. At the lateral boundaries with Dirichlet condition, however, the fluxes do not gain increased accuracy with higher-order polynomials in time. Motivated by this concern, we focus here on 1D elastodynamics and how to treat the Dirichlet boundaries. We introduce two new stabilization methods and show, with test computations, how the stabilized equations used in computing the fluxes perform. The methods are consistent in the way the fluxes are computed and can be extended to 2D and 3D problems.
In this paper, we study asymptotic consensus and finite-time reachability to Pareto optimal set for time-discrete multi-objective consensus-based optimization (M-CBO) algorithm in particle regime without resorting to the mean-field approximation. While the original CBO algorithm [J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci. 28 (2018) 1037–1066] assumes that all particles share common single objective function, the M-CBO algorithm assigns each particle to a distinct sub-objective function obtained by the method of scalarization. In this paper, we show that for a sufficiently small noise, the expected value of the state diameter decays to zero exponentially fast. This results in almost sure asymptotic consensus. We also show that the CBO particle system approximates a Pareto optimal set in finite time. Moreover, the approximation becomes increasingly accurate, as the noise strength vanishes, while the number of particles and the inverse temperature parameter tend to infinity. These results extend our theoretical understanding for CBO-type algorithms to the multi-objective regime by exhibiting asymptotic behaviors that are not observed in the previous mean-field analysis for M-CBO in [G. Borghi, M. Herty and L. Pareschi, An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation, Appl. Math. Optim. 88 (2023) 58].
This paper is concerned with a mass-conservative multiscale finite element framework for single-phase flow in highly heterogeneous porous media. The method combines the local mass conservation and low computational cost of the Enriched Galerkin method within the Generalized Multiscale Finite Element Method (GMsFEM) framework, resulting in the Generalized Multiscale Enriched Galerkin (GMsEG) method. The GMsEG method embeds the fine-scale heterogeneity into the continuous multiscale basis functions by solving local eigen-problems, and the continuous multiscale finite element space is enriched by the piecewise constant space on the coarse grid. The resulting method preserves the mass conservation property and can resolve fine-grid heterogeneity while remaining computationally efficient. Well-posedness and convergence estimates of the method are established, and numerical experiments in various heterogeneous porous media demonstrate the efficiency and accuracy of the method, as well as highlight the significance of employing a mass conservative method when flow and transport problems are coupled together.
This editorial paper introduces the articles published in a special issue focused on essays and reviews, which offer a forward-looking perspectives on new frontiers of applied mathematics. First, a brief description of the scientific contributions of the papers published in this issue is provided. Then, based on the papers’ contents, a forward look at research perspectives is brought to the reader’s attention. These perspectives focus on new developments of the kinetic theory of classical and active particles, behavioral swarms, and numerical and analytical study of hyperbolic systems. Finally, our editorial explores how the contents of this issue can have an impact on mathematical tools of Scientific Machine learning Applied to the study of collective dynamics of living systems.
We establish the local-in-time well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters in 1D derived rigorously from incompressible Navier–Stokes system with a moving free surface by Gerbeau and Perthame. Our solutions are uniformly smooth up to the moving boundary, although the depth degenerates as a singularity of the distance to the vacuum boundary. The main proof is built on some elaborate higher-order weighted energy functional and weighted estimates associated to the degenerate structure of the momentum equation.
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 u(t) = Delta u-del & centerdot; (uS del v); 0 = Delta v-v + u in smoothly bounded planar domains, where S is a matrix attaining value in R-2 & times;2 as S = (cos theta(sin theta) -sin theta(cos theta)) with theta is an element of (- pi/2 , pi/ 2). We shall investigate the system accompanied by boundary conditions of no-flux type for u and of Dirichlet type for v. It is shown that the associated initial-boundary value problem possesses a critical value of the initial mass m(c)= 8 pi/cos theta to distinguish global existence and finite-time blowup. Furthermore, we study the blowup mechanism, and f prove that the number of blowup points is finite and does not exceed cos theta/8 pi integral(Omega) u(0). Our results present a precise characterization of the effect of rotational trajectory on the properties of Keller-Segel system in the form of rotational angle.
Social behaviors play a crucial role in shaping wealth distribution in societies, while wealth distribution also significantly influences 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 kinetic theory for active particles, considering a complex socio-economic system in which individuals are represented as active particles characterized by a vector-valued state encompassing their wealth level and social behavioral preference - cooperation and competition. Here, competition reflects self-interested behavior that resists redistribution and reinforces wealth disparities; In contrast, cooperation is interpreted not as inherent altruism but as consensus behavior: an individual's positive attitude toward redistributive policies such as taxation and welfare that transfer resources to poorer members of society. A multiscale approach is employed, accounting for both decentralized micro-micro (m-m) interactions between individuals and regulatory micro-macro (m-M) interactions between individuals and societal-scale indicators. Numerical simulations reveal that a system with only m-m interactions tends toward polarization, exhibiting "winner-take-all" dynamics that often appear in an unregulated competitive society. In contrast, the m-M interactions, interpreted as macroscopic policy or normative control that enforces consensus, steer the system toward a more stable and balanced economic structure. This work extends kinetic socio-economic models by coupling wealth and behavioral dynamics through a co-evolutionary feedback loop, providing a useful tool for analyzing the impact of policy and social design on economic inequality.
In this work, we propose and analyze a mixed virtual element method within a Banach space framework to numerically investigate the unsteady motion of non-Newtonian pseudoplastic Stokes flows. Motivated by the growing interest in non-Newtonian fluid problems where the stress response plays a pivotal role, our formulation introduces additional unknowns, including the rate of strain and the symmetric stress tensor. This leads to a mixed variational formulation that incorporates the velocity, strain rate, and stress within a Banach space setting. We establish the well-posedness of the weak solution and derive stability estimates using classical results from the theory of nonlinear monotone operators. The discretization in both space and time is performed using an H(div)-conforming virtual element method and the implicit Euler scheme, respectively. Specifically, the pseudostress is approximated using a virtual element subspace of H(div; Omega), while piecewise polynomial subspaces of degree j are employed for approximating the velocity and the rate of strain tensor. The nonlinear term is treated implicitly in the time discretization, and the resulting scheme is shown to be well-posed and unconditionally stable. A rigorous convergence analysis is provided for all unknowns in their natural norms, establishing optimal convergence rates with respect to both the spatial mesh size and the time step. Finally, a series of numerical experiments is presented to validate the accuracy and effectiveness of the proposed method.
The principle of conservation of energy implies that a crack growing in a visco elastic 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 visco elastic body, like the frequently used Kelvin-Voigt model, imply that the dynamic energy-dissipation balance prevents crack growth. This unrealistic result forces us to use different models to describe the evolution of a visco elastic body in the context of dynamic fracture mechanics. In this paper, we consider an example of dynamic visco elastic problem with memory in a two-dimensional domain with a crack growing with constant velocity along a straight line. Through a careful analysis of the singularity of the solutions around the crack tip we show that for suitable values of the material constants there exist solutions that satisfy the energy-dissipation balance. This may suggest that we should use suitable models with memory in the context of dynamic fracture mechanics of visco elastic bodies.
The Random Batch Method (RBM) [S. Jin, L. Li and J.-G. Liu, Random Batch Methods (RBM) for interacting particle systems, J. Comput. Phys. 400 (2020) 108877] is not only an efficient algorithm for simulating interacting particle systems, but also a randomly switching networked model for interacting particle system. This work investigates two RBM variants (RBM-r and RBM-1) applied to the Cucker-Smale flocking model. We establish the asymptotic emergence of global flocking and derive corresponding error estimates. By introducing a crucial auxiliary system and leveraging the intrinsic characteristics of the Cucker-Smale model, and under suitable conditions on the force, our estimates are uniform in both time and particle numbers. In the case of RBM-1, our estimates are sharper than those in [S.-Y. Ha, S. Jin, D. Kim and D. Ko, Uniform-in-time error estimate of the random batch method for the Cucker-Smale model, Math. Models Methods Appl. Sci. 31 (2021) 1099-1135.] Additionally, we provide numerical simulations to validate our analytical results.
We investigate asymptotic stability of constant equilibrium states for compressible non-isothermal cohesive fluids also termed capillary fluids or diffuse interface fluids. The density gradient is added as an extra variable and the augmented system of equations is recast into a normal form with symmetric transport first-order terms, symmetric dissipative second-order terms and antisymmetric cohesive second-order terms. Global existence and asymptotic stability of constant equilibrium states are established by using new dissipative conditions for such augmented hyperbolic-parabolic-dispersive systems of equations. Decay estimates are obtained in all spatial dimensions by using the augmented formulation as well as estimates in Fourier spaces.
This editorial introduces a special issue devoted to recent developments in scientific machine learning. The issue brings together contributions that illustrate, from complementary viewpoints, how mathematical analysis, modeling, scientific computing, control, and learning are becoming increasingly intertwined in the study of complex systems. Beyond presenting the papers collected in this volume, the editorial aims to place them within a broader conceptual landscape and to highlight a number of emerging directions that, in our view, will shape the future of the field. The presentation is organized in three parts: first, the motivations and scientific context underlying the special issue, second, a brief overview of the contributions included in the volume, and third, a discussion of some challenging perspectives at the interface of mathematics, scientific computing, and artificial intelligence.
This work is dedicated to the study, modeling and simulation of the collective dynamics of interacting living system. The core mathematical contribution is a coupled kinetic system for the distribution function f and the learned distribution phi, where social learning and decision-making interact bidirectionally through physics-brain-guided operators. The first objective is to develop some perspective ideas toward the approach of theoretical tools of Artificial Intelligence methods, specifically Scientific Machine Learning, with focus on the study of the collective dynamics of several interacting entities. The study will be developed specifically focusing on the collective behavior of living entities, real or artificial, e.g. imitated by robots, with application to the dynamics of different situations of social life. The second perspective is to develop the conceptual tools for a theory of artificial intelligence. The aim is to model a dynamic in which interacting entities learn from other entities as well as from the environment and external actions. Then, out of this collective social-learning process, each entity develops a strategy to pursue specific goals through a decision-making process that leads to the dynamics. The approach is based on developments in the kinetic theory of active particles. This paper does not naively claim that the problem of artificial intelligence for collective dynamics has been considered exhaustively, but some hints are proposed to contribute to such a challenging perspective in view of further developments.
This paper deals with the following chemotaxis system with gradient-dependent flux limitation and nonlinear diffusion {u(t) = del & sdot; (D(u)del u) -del & sdot; (u(1 + |del v|(2))(-alpha/2)del v), x is an element of Omega, t > 0, 0 = Delta v - mu + u, x is an element of Omega, t > 0 under homogeneous Neumann boundary conditions in a smoothly bounded domain Omega subset of & Ropf;(n) (n >= 1), where alpha > 0, mu = 1/|Omega|integral Omega udx, D generalizes the prototype given by D(zeta) = (zeta + iota)(m-1) for all zeta >= 0 with iota > 0, m > 0. It is shown that the corresponding initial-boundary value problem possesses a unique globally bounded classical solution provided that alpha > 2 - nm/n-1 and m > 1 - 1/n. While for Omega = B-R(0) subset of & Ropf;(n) and 0 < alpha < min{2 - nm/n-1, n-2/n-1}, there exist nonnegative radially symmetric initial data such that the solution blows up in finite time. These conditions are optimal for m >= 1, especially when m = 1, which is consistent with the results in Winkler [Indiana Univ. Math. J. 71 (2022) 1437-1465]. Furthermore, along the critical line alpha = 2-nm/n-1 with m >= 1, the finite-time blow-up of radial solutions with sufficiently large initial mass is also established, conversely, the global boundedness of solutions with suitably small initial data is proved.
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 shape their ecological niche. To this end, we adapt the Kinetic Theory of Active Particles to model virus interactions, allowing payoffs to co-evolve as a function of the population's configuration. We deduce the system of ordinary equations corresponding to the replicator dynamics with frequency-dependent payoffs. Then we obtain a nonlinear integro-differential equation describing the dynamics for a continuum of strategies by passing to the limit in the replicator system when the number of equations grows. Finally, we present some examples of virus dynamics.
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 applications, where mathematics interacts with natural sciences, social systems and economics. A key aim of the content of this issue is the perspective of understanding how mathematics can contribute to Scientific Machine Learning toward Artificial Intelligence.
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 complexity by restricting communication to dynamically formed batches of the whole swarm. We provide several sufficient frameworks for stochastic consensus in noise-free and noisy environments, that is, under mild assumptions on system parameters, the swarm achieves almost sure alignment in both activity and heading angle, while maintaining bounded spatial dispersion. Numerical simulations validate the theoretical findings, illustrating that random batch interactions yield consensus behaviors comparable to all-to-all communication while significantly reducing computational cost. The results provide a scalable framework for analyzing and simulating large-scale swarm systems with applications to the modeling of collective behaviors and decentralized controls.
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 shaping infection dynamics, with a focus on virus recognition and immune system activation. Simulations show that this approach aligns with clinical observations such as the dual role of interferons in antiviral defence and immunopathology. This highlights the potential of the model to inform therapeutic strategies.
In this work, we study the doubly degenerate nutrient taxis system { u(t )= del & sdot; (uv del u) - chi del & sdot; (u(2)v del v) + & ell;uv, v(t) = Delta v - uv in a smoothly bounded domain Omega subset of & Ropf;(2), where chi > 0 and & ell; > 0. This model was proposed by [J. F. Leyva, C. M & aacute;laga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with nonlinear degenerate cross diffusion, Phys. A 392 (2013) 5644-5662] to describe the experimentally observed complex pattern formation phenomena in bacterial populations. In this paper, we demonstrate that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem (star) possesses a global bounded weak solution, which is continuous in its first component and essentially smooth in its second component. Compared to the existing works, we do not impose any smallness conditions on the initial data. Moreover, we identify a criterion regarding the initial smallness of the second component such that the solution stabilizes to a nonconstant steady state.
Working with the one-dimensional Schr & ouml;dinger equation in impedance form, we derive an exact inverse scattering formula that expresses impedance in terms of the reflection coefficient, and we prove injectivity of the scattering map for impedance functions of lower regularity than previously analyzed. The inverse scattering formula translates directly into an efficient numerical algorithm that accurately transforms digital scattering data into impedance. The results apply to acoustic imaging of layered media, as well as to inverse quantum scattering.
