Mathematical Models and Methods in Applied Sciences最新文献

{"title":"Unifying local and nonlocal corrosion frameworks: A convergent nonlocal extension of the KKS phase-field model","authors":"Alexander Hermann,Marvin Fritz,Tobias Koppl,Arman Shojaei,Stewart A. Silling,Christian J. Cyron","doi":"10.1142/s0218202526500594","DOIUrl":"https://doi.org/10.1142/s0218202526500594","url":null,"abstract":"We introduce a nonlocal extension of the Kim-Kim-Suzuki (KKS) phase-field corrosion model aimed at bridging local and nonlocal corrosion modeling approaches, such as phase-field and peridynamic frameworks. In this formulation, classical gradient operators are replaced with integral operators defined over a finite interaction horizon, naturally embedding an intrinsic length scale that aligns with nonlocal theories like peridynamics. Under precise assumptions on function spaces and kernel functions, we define a nonlocal free energy that integrates a standard bulk free energy density with a nonlocal interaction term. Through differentiation in an appropriate Hilbert space, we derive evolution equations, yielding a nonlocal Allen–Cahn equation for the phase-field and a nonlocal Cahn–Hilliard-type equation for the concentration. The latter is expressed as a gradient flow in a metric induced by the inverse of the nonlocal operator, mirroring the classical H −1 metric for conserved dynamics. We establish the well-posedness of these equations using Galerkin approximations, uniform energy estimates, and compactness arguments. Furthermore, we prove the convergence of the nonlocal model to its local KKS counterpart as the interaction horizon approaches zero, effectively unifying local and nonlocal perspectives. Numerical experiments, implemented via an explicit, matrix-free discretization, confirm the asymptotic compatibility of the scheme as the horizon vanishes and illustrate the transition from diffuse to sharp corrosion fronts. We further demonstrate the framework on a three-dimensional biodegradable magnesium implant screw, calibrated against experimental degradation data, and show that its explicit, matrix-free structure scales efficiently across compute nodes and onto GPUs, making engineering-scale simulations tractable.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"385 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149208867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a Mullins–Sekerka model for the growth of active droplets modelling protocells: Stability analysis and numerical computations","authors":"Harald Garcke,Kei Fong Lam,Robert Nurnberg,Andrea Signori","doi":"10.1142/s0218202526500600","DOIUrl":"https://doi.org/10.1142/s0218202526500600","url":null,"abstract":"Mullins–Sekerka models with chemical reactions can lead to scenarios where droplets grow, become unstable, split, grow and undergo further division. These grow and division cycles have been proposed as a model for protocells and are believed to play a fundamental role in living systems by providing chemical compartments which are important in the organization of living systems. This paper analyses chemically active Mullins–Sekerka models. Existence of radially symmetric solutions is shown and a detailed stability analysis in radial as well as planar situations is given. In particular, we also analyze multilayered solutions leading to shell-type situations. Finally, we introduce a numerical method based on a parametric finite element approach that explicitly accounts for topological changes, thereby allowing for droplet splitting and merging. Several numerical simulations verify the findings of the theoretical stability analysis and show complex dynamical behavior, including multiple instabilities, splittings of droplets and appearance of shell-type solutions.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149208866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deep neural networks and finite elements of any order on arbitrary dimensions","authors":"Juncai He,Jinchao Xu","doi":"10.1142/s0218202527500047","DOIUrl":"https://doi.org/10.1142/s0218202527500047","url":null,"abstract":"In this study, we establish that deep neural networks employing ReLU and ReLU 2 activation functions can rigorously represent Lagrange finite element functions of any order on arbitrary simplicial meshes in any dimension. We introduce two novel formulations for globally expressing the basis functions of Lagrange elements, tailored for both specific and arbitrary meshes. These formulations are based on a geometric decomposition of the elements, incorporating several insightful and essential properties of high-dimensional simplicial meshes, barycentric coordinate functions, and global basis functions of linear elements. This representation theory facilitates a natural approximation result for such deep neural networks. Our findings present the first demonstration of how deep neural networks can systematically generate general continuous piecewise polynomial functions on both specific and arbitrary simplicial meshes.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"2 1","pages":"1-25"},"PeriodicalIF":0.0,"publicationDate":"2026-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149020995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Provably positivity-preserving, globally divergence-free central DG methods for ideal MHD system","authors":"Ruifang Yan,Huihui Cao,Kailiang Wu","doi":"10.1142/s0218202527500059","DOIUrl":"https://doi.org/10.1142/s0218202527500059","url":null,"abstract":"The compressible MHD equations possess two essential structural properties: (i) an algebraic structure ensuring the positivity of density and pressure; and (ii) a differential structure maintaining the divergence-free (DF) constraint on the magnetic field. A deep connection between these two properties has been revealed in both non-central [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124–2147] and central discontinuous Galerkin (CDG) frameworks [K. Wu, H. Jiang & C.-W. Shu, SIAM J. Numer. Anal., 61 (2023), pp. 250–285]. However, existing methods could provably preserve positivity only in conjunction with a locally DF property. Constructing a uniformly high-order method that is both provably positive and globally DF has remained an open and challenging problem. This paper proposes a numerical method, termed PosDiv-CDG, that provably preserves both positivity and the global DF condition at arbitrarily high order in multiple dimensions. It resolves the fundamental structural incompatibility between standard positivity-preserving limiters and global DF enforcement in the CDG framework. The method integrates a novel positivity-limiting strategy, a modified dissipation mechanism guided by convex decomposition, and an auxiliary evolution equation for the magnetic field, which are designed based on rigorous theoretical analysis. Notably, we provide a rigorous proof of positivity preservation for the updated auxiliary full-state averages under an explicit CFL-type condition. The proof leverages the geometric quasi-linearization (GQL) technique, which reformulates the nonlinear positivity constraint into an equivalent linear form. This enables the derivation of flux-based inequalities and technical estimates under the global DF constraint. To suppress nonphysical oscillations near shocks, we develop a compact, non-intrusive convex-oscillation-suppressing (COS) procedure based on the entropy function. The COS process acts only on non-magnetic variables, avoids costly characteristic decomposition, and maintains both the globally DF property and high-order accuracy. Several challenging experiments—including low plasma-beta MHD jets with Mach numbers up to 1,000,000—demonstrate the proposed method robustness, high-order accuracy, non-oscillatory behavior, and its ability to preserve both positivity and globally DF structures under extreme conditions.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149020996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiphase Cross-diffusion models for tissue structures: modeling, analysis, numerics","authors":"Ansgar Jungel,Cordula Reisch,Sara Xhahysa","doi":"10.1142/s0218202527500035","DOIUrl":"https://doi.org/10.1142/s0218202527500035","url":null,"abstract":"Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell–Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak–strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149020997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Isogeometric\u0000 C\u0000 1\u0000 mortar method","authors":"A. Benvenuti,G. Loli,G. Sangalli,T. Takacs","doi":"10.1142/s0218202527500060","DOIUrl":"https://doi.org/10.1142/s0218202527500060","url":null,"abstract":"We present an isogeometric mortar method for the discretization of the biharmonic equation posed on multi-patch domains. We assume only C 0 -conformity at interfaces and employ a mortar approach to weakly enforce C 1 -continuity across patch interfaces. Discrete inf-sup stability is ensured by selecting a Lagrange multiplier space consisting of splines of degree reduced by two compared to the primal space, with increased smoothness or merged elements near vertices. We prove optimal a priori error estimates and confirm the theoretical findings with a series of numerical experiments.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"106 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149020993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pulsatile Flows for Simplified Smart Fluids with Variable Exponent Power-Law: Analysis and Numerics","authors":"Luigi C. Berselli,Alex Kaltenbach","doi":"10.1142/s0218202527500023","DOIUrl":"https://doi.org/10.1142/s0218202527500023","url":null,"abstract":"We study the fully-developed, time-periodic motion of a shear-dependent non-Newtonian fluid with variable exponent rheology through an infinite pipe [Formula: see text], [Formula: see text], of arbitrary cross-section [Formula: see text]. The focus is on a generalized [Formula: see text]-fluid model, where the power-law index is position-dependent (with respect to [Formula: see text]), i.e., a function [Formula: see text]. We prove the existence of time-periodic solutions with either assigned time-periodic flow-rate or pressure-drop, generalizing known results for the Navier–Stokes and for [Formula: see text]-fluid equations. In addition, we identify explicit solutions, relevant as benchmark cases, especially for electro-rheological fluids or, more generally, ‘smart fluids’. To support practical applications, we present a fully-constructive existence proof for variational solutions by means of a fully-discrete finite-difference/-element discretization, consistent with our numerical experiments. Our approach, which unifies the treatment of all values of [Formula: see text], [Formula: see text], without requiring an auxiliary Newtonian term, provides new insights even in the constant exponent case. The theoretical findings are illustrated by means of numerical experiments.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149020994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Continuous-Time Heterogeneous Agent Models with Recursive Utility and Preference for Late Resolution","authors":"Yves Achdou,Qing Tang","doi":"10.1142/s0218202527500011","DOIUrl":"https://doi.org/10.1142/s0218202527500011","url":null,"abstract":"We consider continuous-time heterogeneous agent models with recursive utility (Epstein-Zin utility) cast as mean field games, in which agents prefer late resolution of uncertainty. The model leads to a system coupling a pair of Hamilton-Jacobi-Bellman equations with state constraints and Fokker-Planck-Kolmogorov equations. We investigate the existence of solutions to the mean field game system and discuss some important qualitative features of the model.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149020999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Low-regularity integrator and inviscid limit problem for the KdV–Burgers equation","authors":"Xintong Li,Yongsheng Li","doi":"10.1142/s0218202526500545","DOIUrl":"https://doi.org/10.1142/s0218202526500545","url":null,"abstract":"This paper investigates the inviscid limit and time discretizations for the Korteweg–de Vries–Burgers (KdV–B) equation on a torus, focusing on low-regularity solutions below [Formula: see text]. First, we establish an [Formula: see text] error estimate between the exact solutions of the KdV–B and the KdV equations, achieving a convergence rate of order [Formula: see text]. Second, for numerical approximations, we propose a scheme that is uniformly convergent with respect to the diffusion coefficient [Formula: see text]. Under the regularity assumption [Formula: see text] for [Formula: see text], we prove that the proposed scheme achieves a convergence rate of order [Formula: see text] (up to a logarithmic factor) in [Formula: see text], with an error constant that is independent of [Formula: see text]. Consequently, an error bound is derived between the numerical solution of the KdV–B equation and that of the standard KdV equation. Extensive numerical experiments, including comparisons of the proposed method with classical schemes, validate our theoretical results.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"13 1","pages":"1-59"},"PeriodicalIF":0.0,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148894109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A nonlocal equation describing tumor growth","authors":"Rafael Granero-Belinchón, Martina Magliocca","doi":"10.1142/s0218202525500058","DOIUrl":"https://doi.org/10.1142/s0218202525500058","url":null,"abstract":"Cancer is a very complex phenomenon that involves many different scales and situations. In this paper, we consider a free boundary problem describing the evolution of a tumor colony and we derive a new asymptotic model for tumor growth. We focus on the case of a single phase tumor colony taking into account chemotactic effects in an early stage where there is no necrotic inner region. Thus, our model is valid for the case of multilayer avascular tumors with very little access to both nutrients and inhibitors or the case where the amount of nutrients and inhibitors is very similar to the amount consumed by the multilayer tumor cells. Our model takes the form of a single nonlocal and nonlinear partial differential equation for the interface of the multilayer tumor colony. Our model is able to capture chemotactic and cohesion effects and also the effect of nutrients, inhibitors and vasculation of the tumor colony. Besides deriving the model, we also prove a well-posedness result.","PeriodicalId":18311,"journal":{"name":"Mathematical Models and Methods in Applied Sciences","volume":"35 03","pages":"585-609"},"PeriodicalIF":0.0,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147911909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}