Annals of Applied Mathematics
type: channel
description: Annals of Applied Mathematics (AAM) was established in 1985, with the original title Annals of Differential Equations. AAM is an international journal devoted to the publication of original research papers and review articles in all areas of applied mathematics, with an emphasis on differential equations, probability and statistics, computational mathematics and discrete mathematics. The journal is published in one volume annually consisting of four issues. AAM is co-sponsored by Fuzhou University and China Society for Industrial and Applied Mathematics (CSIAM).
publisher: Global Science Press
language: en
publication name: Annals of Applied Mathematics
issn: 2096-0174
items: On Residual Minimization for PDEs: Failure of PINN, Modified Equation, and Implicit Bias
items: Multidimensional Simple Wave and Monge-Ampère Equation: Theory and Applications
items: A Note on Hypergraph Extensions of Mantel’s Theorem
items: The 3-Coloring of Planar Graphs with Adjacent Triangles
items: A Survey on the Performance of Krylov Subspace Methods in High Order Compact Schemes for Solving Poisson’s Equation for Application in Incompressible Fluid Flow Solvers
items: The Method of Fundamental Solutions for Optical Fluorescence
items: Stability and Convergence Analysis of a Linear Energy Stable Scheme for a Cahn-Hilliard Model with Smooth or Weakly Singular Non-Local Term
items: A Hybrid Conjugate Gradient Method with Trust Region for Large-Scale Unconstrained Optimization Problems
items: Numerical Approaches to Compute Spectra of Non-Self Adjoint Operators in Two and Three Dimensions
items: An Extension of the $r^p$ Method for Wave Equations with Scale-Critical Potentials and First-Order Terms
items: Deep Neural Network Approaches for Computing the Defocusing Action Ground State of Nonlinear Schrödinger Equation
items: The $H^p-H^q$ Estimates for a Class of Dispersive Equations with Finite Type Geometry
items: Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three
items: Preface: Special Issue Dedicated to Professor Avy Soffer’s 70th Birthday (Part II)
items: Oscillation-Eliminating Discontinuous Galerkin Methods for Multicomponent Chemically Reacting Flows
items: Reconfiguration Graphs for Vertex Colorings of $P_5$-Free Graphs
items: An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media
items: Existence of Global Attractor for Weakly Damped FDS Nonlinear Wave Equations
items: The Characteristics of Solutions to Semilinear Wave Equation with Logarithmic Plus Polynomial Nonlinearities
items: Non-Global Existence of Regular Solution to Initial Value Problem of Relativistic Euler Equations in $\mathbb{R}^N$
items: Energy Equality for the Isentropic Compressible Navier-Stokes Equations Without Upper Bound of the Density
items: Global Classical Solution and Asymptotic Behavior to a Kind of Linearly Degenerate Quasilinear Hyperbolic System
items: Preface: Special Issue Dedicated to Professor Avy Soffer’s 70th Birthday (Part I)
items: Entire Sign-Changing Solutions to the Fractional Critical Schrödinger Equation
items: On the Monotonicity of $Q^3$ Spectral Element Method for Laplacian
items: Numerical Study of Semidiscrete Penalty Approach for Stabilizing Boussinesq System with Localized Feedback Control
items: On a Hybrid Method for Inverse Transmission Eigenvalue Problems
items: Generalized Isospectral-Nonisospectral Modified Korteweg-de Vries Integrable Hierarchies and Related Properties
items: Empirical Study on Option Pricing under Markov Regime Switching Economics
items: Error Analysis of the Nonconforming $P_1$ Finite Element Method to the Sequential Regularization Formulation for Unsteady Navier-Stokes Equations

On Residual Minimization for PDEs: Failure of PINN, Modified Equation, and Implicit Bias
type: item
description: As a popular and easy-to-implement machine learning method for solving differential equations, the physics-informed neural network (PINN) sometimes may fail and find poor solutions which bias against the exact ones. In this paper, we establish a framework of modified equation to explain the failure phenomenon and characterize the implicit bias of a general residual minimization (RM) method. We provide a simple way to derive the modified equation which models the numerical solution obtained by RM methods. Next, we show the modified solution deviates from the original exact solution. The proof uses a by-product of this paper, that is, a necessary and sufficient condition on characterizing the singularity of the coefficients. This equivalent condition can be extended to other types of equations in the future. Finally, we prove, as a complete characterization of the implicit bias, that RM method implicitly biases the numerical solution against the exact solution and towards a modified solution. In this work, we focus on elliptic equations with discontinuous coefficients, but our approach can be extended to other types of equations and our understanding of the implicit bias may shed light on further development of deep learning based methods for solving equations.
subject: Articles
  topic: section
subject: residual minimization
  topic: keywords
subject: PINN
  topic: keywords
subject: implicit bias
  topic: keywords
subject: modified equation
  topic: keywords
creator: Tao Luo
creator: Qixuan Zhou
rights: Copyright (c) 2026 Annals of Applied Mathematics
date: 2026-01-21
publication date: 2026-01-21
starting page: 313
ending page: 376
doi: 10.4208/aam.OA-2025-0028

Multidimensional Simple Wave and Monge-Ampère Equation: Theory and Applications
type: item
description: This paper explores the properties and mathematical formulations of multidimensional simple waves, extending the well-established theory of one-dimensional simple waves to higher dimensions. The study focuses on the connection between simple waves and the Monge-Ampère equation, particularly in the context of gas dynamics and potential flows. Key aspects include the characterization of simple waves in unsteady and steady flows, the role of characteristic lines, and the application of Hodograph and Legendre transformations to derive solutions. The paper also addresses the challenges and open questions in extending simple wave theory to more complex systems, such as non-reducible systems, radiative heat transfer, and chemical reactions. The research highlights both theoretical advancements and practical applications, providing a foundation for future studies in this area.
subject: Articles
  topic: section
subject: Simple wave
  topic: keywords
subject: Monge-Ampère equation
  topic: keywords
subject: Gas dynamics
  topic: keywords
subject: Characteristic lines
  topic: keywords
subject: Hodograph transformation
  topic: keywords
creator: Boling Guo
creator: Yueyang Feng
creator: Yaqing Liu
rights: Copyright (c) 2026 Annals of Applied Mathematics
date: 2026-01-21
publication date: 2026-01-21
starting page: 267
ending page: 294
doi: 10.4208/aam.OA-2025-0027

A Note on Hypergraph Extensions of Mantel’s Theorem
type: item
description: Chao and Yu introduced an entropy method for hypergraph Turán problems, and used it to show that the family of $⌊k/2⌋$ $k$-uniform tents have Turán density $k!/{k^k}$ . Il’kovič and Yan [5] improved this by reducing to a subfamily of ${\lceil}k/e⌉$ tents. In this note, enhancing Il’kovič-Yan’s result, we give a significantly shorter entropy proof, with optimal bounds within this framework.
subject: Articles
  topic: section
subject: Turán density
  topic: keywords
subject: entropy
  topic: keywords
subject: hypergraph
  topic: keywords
creator: Jie Ma
creator: Tianming Zhu
rights: Copyright (c) 2026 Annals of Applied Mathematics
date: 2026-01-21
publication date: 2026-01-21
starting page: 295
ending page: 301
doi: 10.4208/aam.OA-2025-0016

The 3-Coloring of Planar Graphs with Adjacent Triangles
type: item
description: Erdős raised the following problem according to Steinberg’s conjecture: Is there an integer such that every planar graph without cycles of length from 4 to $k$ is 3-colorable. By far, the result about the problem was improved to $k≤7$ by Borodin et al. However, by permitting the existence of adjacent triangles except $K_4$, for an arbitrary integer $k≥5,$ there exists a planar graph without cycles of length from 5 to $k$ such that $G$ is not 3-colorable. Let $d$ denote the minimum distance between two diamonds in $G$, where a diamond is the union of two adjacent triangles. In this paper, we prove that a planar graph $G$ with $d≥2$ and without cycles of length from 5 to 18 is 3-colorable. The reader is invited to find the smallest integer $k$ such that a planar graph $G$ with $d≥2$ and without cycles of length from 5 to $k$ is 3-colorable.
subject: Articles
  topic: section
subject: Three Color Problem
  topic: keywords
subject: adjacent triangles
  topic: keywords
subject: cycle
  topic: keywords
subject: planar graph
  topic: keywords
creator: Zuosong Liang
creator: Xin Xue
rights: Copyright (c) 2026 Annals of Applied Mathematics
date: 2026-01-21
publication date: 2026-01-21
starting page: 302
ending page: 312
doi: 10.4208/aam.OA-2025-0013

A Survey on the Performance of Krylov Subspace Methods in High Order Compact Schemes for Solving Poisson’s Equation for Application in Incompressible Fluid Flow Solvers
type: item
description: The efficiency of three Krylov subspace methods with their ILU0-preconditioned version in solving the systems with the nondiagonal sparse matrix is examined. The systems have arisen from the discretization of Poisson’s equation using the 4th and 6th-order compact schemes. Four matrix-vector multiplication techniques based on four sparse matrix storage schemes are considered in the algorithm of the Krylov subspace methods and their effects are explored. The convergence history, error reduction, iteration-resolution relation and CPU-time are addressed. The efficacy of various methods is evaluated against a benchmark scenario in which the conventional second-order central difference scheme is employed to discretize Poisson’s equation. The Krylov subspace methods, paired with four distinct matrix-vector multiplication strategies across three discretization approaches, are tested and implemented within an incompressible fluid flow solver to solve the elliptic segment of the equations. The resulting solution process CPU-time surface gives a new vision regarding speeding up a CFD code with proper selection of discretization stencil and matrix-vector multiplication technique.
subject: Articles
  topic: section
subject: High order compact
  topic: keywords
subject: Krylov subspace methods
  topic: keywords
subject: Navier-Stokes equations
  topic: keywords
subject: Poisson’s equation
  topic: keywords
subject: CPU-time
  topic: keywords
subject: matrix-vector multiplication
  topic: keywords
subject: sparse storage schemes.
  topic: keywords
subject: 65F10
  topic: subjects
subject: 76D05
  topic: subjects
creator: Iman Farahbakhsh
creator: Benyamin Barani Nia
creator: Mehdi Dehghan
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-04
publication date: 2025-06-04
starting page: 239
ending page: 266
doi: 10.4208/aam.OA-2024-0025

The Method of Fundamental Solutions for Optical Fluorescence
type: item
description: In this paper, we develop the method of fundamental solutions (MFS) for solving boundary value problems in the field of optical fluorescence. The governing system of diffusion-absorption equations for the excitation and emission fluences is transformed into a single fourth-order partial differential equation whose fundamental solution can be expressed as the difference of two fundamental solutions of the complex Helmholtz equation. The numerically obtained results confirm the accuracy of the MFS when compared with an available analytical solution. Numerical results are also provided for a physical application in optical fluorescence. Furthermore, extensions to three dimensions along with numerical verification are performed.
subject: Articles
  topic: section
subject: Optical fluorescence
  topic: keywords
subject: method of fundamental solutions
  topic: keywords
subject: elliptic equations.
  topic: keywords
subject: 65N80
  topic: subjects
subject: 65N38
  topic: subjects
creator: Andreas Karageorghis
creator: Daniel Lesnic
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-04
publication date: 2025-06-04
starting page: 219
ending page: 238
doi: 10.4208/aam.OA-2025-0010

Stability and Convergence Analysis of a Linear Energy Stable Scheme for a Cahn-Hilliard Model with Smooth or Weakly Singular Non-Local Term
type: item
description: We consider a Cahn-Hilliard gradient flow model with a free energy functional, which contains a non-local term in addition to linear and non-linear local terms. The non-local terms can be based on smooth and weakly singular kernel operators. We establish the well-posedness of this problem, construct an unconditional energy stable scheme, and carry out a stability and convergence analysis. Several numerical results are presented to illustrate the efficiency and robustness of the proposed scheme.
subject: Articles
  topic: section
subject: Cahn-Hilliard
  topic: keywords
subject: weakly singular
  topic: keywords
subject: non-local
  topic: keywords
subject: energy stable
  topic: keywords
subject: existence and uniqueness
  topic: keywords
subject: stability and convergence.
  topic: keywords
subject: 65M12
  topic: subjects
subject: 65M70
  topic: subjects
subject: 35K35
  topic: subjects
subject: 35K61
  topic: subjects
creator: Moumita Mandal
creator: Manisha Chowdhury
creator: Jie Shen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-04
publication date: 2025-06-04
starting page: 194
ending page: 218
doi: 10.4208/aam.OA-2025-0004

A Hybrid Conjugate Gradient Method with Trust Region for Large-Scale Unconstrained Optimization Problems
type: item
description: In this work, we modify a conjugate gradient (CG) method recently proposed in the literature, where a PRP conjugate gradient method is modified using trust region. Particularly, we propose a hybrid CG method that incorporates the parameters $β^{PRP},$ $β^{FR}$ and $β^{CD},$ and this new search direction satisfies both the trust region feature and the sufficient descent conditions. Furthermore, under suitable conditions the developed method is proved to be globally convergent. The method is tested on some benchmark problems from the literature and numerical results show that it is quite efficient in solving large scale problems.
subject: Articles
  topic: section
subject: Conjugate gradient method
  topic: keywords
subject: global convergence
  topic: keywords
subject: strong Wolfe line search
  topic: keywords
subject: trust-region.
  topic: keywords
subject: 90C06
  topic: subjects
subject: 90C30
  topic: subjects
subject: 65K05
  topic: subjects
creator: A. P. Byengonzi
creator: P. Kaelo
creator: M. Koorapetse
creator: P. Mtagulwa
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-04
publication date: 2025-06-04
starting page: 176
ending page: 193
doi: 10.4208/aam.OA-2024-0019

Numerical Approaches to Compute Spectra of Non-Self Adjoint Operators in Two and Three Dimensions
type: item
description: In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators, in two and three spatial dimensions. Indeed, in the multidimensional case very few results are known on the location of the eigenvalues. This leads to solve nonlinear eigenvalue problems. In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case. Then we present the numerical methods developed to compute the spectra (finite difference discretization) for the two and three dimensional cases. The numerical results obtained are presented and analyzed. One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable. This work is in continuity of a previous work in one spatial dimension [3].
subject: Articles
  topic: section
subject: Non-self adjoint quadratic operators
  topic: keywords
subject: nonlinear eigenvalue problems
  topic: keywords
subject: spectra
  topic: keywords
subject: finite difference methods.
  topic: keywords
subject: 65F15
  topic: subjects
subject: 65D25
  topic: subjects
creator: Fatima Aboud
creator: François Jauberteau
creator: Didier Robert
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-06-04
publication date: 2025-06-04
starting page: 155
ending page: 175
doi: 10.4208/aam.OA-2024-0026

An Extension of the $r^p$ Method for Wave Equations with Scale-Critical Potentials and First-Order Terms
type: item
description: The $r^p$ method, first introduced in [9], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the $r^p$-weighted energy using a novel Grönwall argument, which allows a larger range of $p$ than the standard method. A spherically symmetric version of our strategy first appeared in [22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.
subject: Articles
  topic: section
subject: Scale-critical potential
  topic: keywords
subject: $r^p$ method.
  topic: keywords
subject: 35
  topic: subjects
creator: Maxime Van de Moortel
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-15
publication date: 2025-04-15
starting page: 112
ending page: 154
doi: 10.4208/aam.OA-2025-0003

Deep Neural Network Approaches for Computing the Defocusing Action Ground State of Nonlinear Schrödinger Equation
type: item
description: The defocusing action ground state of the nonlinear Schrödinger equation can be characterized via three different but equivalent minimization formulations. In this work, we propose some deep neural network (DNN) approaches to compute the action ground state through the three formulations. We first consider the unconstrained formulation, where we propose the DNN with a shift layer and demonstrate its necessity towards finding the correct ground state. The other two formulations involve the $L^{p+1}$-normalization or the Nehari manifold constraint. We enforce them as hard constraints into the networks by further proposing a normalization layer or a projection layer to the DNN. Our DNNs can then be trained in an unconstrained and unsupervised manner. Systematical numerical experiments are conducted to demonstrate the effectiveness and superiority of the approaches.
subject: Articles
  topic: section
subject: Nonlinear Schrödinger equation
  topic: keywords
subject: action ground state
  topic: keywords
subject: deep neural network
  topic: keywords
subject: shift layer
  topic: keywords
subject: normalization layer
  topic: keywords
subject: projection layer.
  topic: keywords
subject: 35Q55
  topic: subjects
subject: 68T07
  topic: subjects
subject: 81-08
  topic: subjects
subject: 81Q05
  topic: subjects
creator: Zhipeng Chang
creator: Zhenye Wen
creator: Xiaofei Zhao
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-15
publication date: 2025-04-15
starting page: 42
ending page: 76
doi: 10.4208/aam.OA-2024-0023

The $H^p-H^q$ Estimates for a Class of Dispersive Equations with Finite Type Geometry
type: item
description: This paper studies the $H^p-H^q$ estimates of a class of oscillatory integrals related to dispersive equations under the assumption that the level hypersurfaces are convex and of finite type. As applications, we obtain the decay estimates for the solutions of higher order homogeneous and inhomogeneous Schrödinger equations.
subject: Articles
  topic: section
subject: Dispersive equations
  topic: keywords
subject: $H^p-H^q$ estimates
  topic: keywords
subject: finite type geometry
  topic: keywords
subject: decay estimates.
  topic: keywords
subject: 42B30
  topic: subjects
subject: 42B37
  topic: subjects
creator: Qingquan Deng
creator: Xuejian Meng
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-15
publication date: 2025-04-15
starting page: 77
ending page: 111
doi: 10.4208/aam.OA-2025-0001

Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three
type: item
description: This paper is concerned with the time decay estimates of the fourth order Schrödinger operator $H = ∆^2+V (x)$ in dimension three, where $V (x)$ is a real valued decaying potential. Assume that zero is a regular point or the first kind resonance of $H,$ and $H$ has no positive eigenvalues, we established the following time optimal decay estimates of $e^{−it H}$ with a regular term $H^{α/4}:$ When zero is the second or third kind resonance of $H,$ their decay will be significantly changed. We remark that such improved time decay estimates with the extra regular term $H^{α/4}$ will be interesting in the well-posedness and scattering of nonlinear fourth order Schrödinger equations with potentials.
subject: Articles
  topic: section
subject: Fourth order Schrödinger equation
  topic: keywords
subject: asymptotic expansions
  topic: keywords
subject: $L^1−L^∞$ decay estimate
  topic: keywords
subject: resonances.
  topic: keywords
subject: 58J50
  topic: subjects
subject: 42B15
  topic: subjects
subject: 35P15
  topic: subjects
subject: 42B20
  topic: subjects
subject: 47F05
  topic: subjects
creator: Ping Li
creator: Zijun Wan
creator: Hua Wang
creator: Xiaohua Yao
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-15
publication date: 2025-04-15
starting page: 1
ending page: 41
doi: 10.4208/aam.OA-2024-0018

Preface: Special Issue Dedicated to Professor Avy Soffer’s 70th Birthday (Part II)
type: item
description: Professor Avraham (Avy) Soffer was born in Israel on December 28, 1955. He earned his Bachelor’s, Master’s, and Doctor’s degrees in Science from Tel-Aviv University in 1976, 1980, and 1984, respectively. Between 1978 and 1983, he worked as an assistant at Tel-Aviv University.
subject: Articles
  topic: section
subject:
  topic: keywords
subject:
  topic: subjects
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-15
publication date: 2025-04-15
starting page: i
ending page: iv
doi: 10.4208/aam.2025.v41n1.preface

Oscillation-Eliminating Discontinuous Galerkin Methods for Multicomponent Chemically Reacting Flows
type: item
description: This paper proposes a robust and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method for solving multicomponent chemically reacting flows, which is an extension and application of the recent work [M. Peng, Z. Sun, and K. Wu, Math. Comput., 2024, doi.org/10.1090/mcom/3998]. Following recently developed high-order bound-preserving discontinuous Galerkin method in [J. Du and Y. Yang, J. Comput. Phys., 469 (2022), 111548], we incorporate an OE procedure after each Runge-Kutta time stage to suppress spurious oscillations. The OE procedure is defined by the solution operator of a damping equation, which can be analytically solved without requiring discretization, making its implementation straightforward, non-intrusive, and efficient. Through careful design of the damping coefficients, the proposed OEDG method not only achieves the essentially non-oscillatory (ENO) property without compromising accuracy but also preserves the conservative property — an indispensable aspect of the bound-preserving technique introduced in [J. Du and Y. Yang, J. Comput. Phys., 469 (2022), 111548]. The effectiveness and robustness of the OEDG method are demonstrated through a series of one- and two-dimensional numerical tests on the compressible Euler and Navier–Stokes equations for chemically reacting flows. These results highlight the method’s capability to handle complex flow dynamics while maintaining stability and high-order accuracy.
subject: Articles
  topic: section
subject: Oscillation elimination discontinuous Galerkin (OEDG) method
  topic: keywords
subject: modal filter
  topic: keywords
subject: bound-preserving
  topic: keywords
subject: multicomponent chemically reacting flows.
  topic: keywords
subject: 65M12
  topic: subjects
subject: 65M60
  topic: subjects
subject: 76N06
  topic: subjects
creator: Jie Du
creator: Chuan Fan
creator: Kailiang Wu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-01-14
publication date: 2025-01-14
starting page: 412
ending page: 444
doi: 10.4208/aam.OA-2024-0029

Reconfiguration Graphs for Vertex Colorings of $P_5$-Free Graphs
type: item
description: For any positive integer $k,$ the reconfiguration graph for all $k$-colorings of a graph $G,$ denoted by $\mathcal{R}_k(G),$ is the graph where vertices represent the $k$-colorings of $G,$ and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any 2-chromatic $P_5$-free graph $G,$ $\mathcal{R}_k(G)$ is connected for each $k≥3.$ On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p,$ such that $\mathcal{R}_{8p}(G)$ is disconnected. In this paper, we offer a detailed classification of the connectivity of $\mathcal{R}_k(G)$ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t = 3,$ and $t ≥ 4$ with $t+1 ≤ k ≤(\begin{matrix} t \\ 2 \end{matrix}).$ We demonstrate that $\mathcal{R}_k(G)$ remains connected for each 3-chromatic $P_5$-free graph $G$ and each $k ≥4.$ Furthermore, for each $t≥4$ and $t+1\le k\le (\begin{matrix} t \\ 2 \end{matrix}),$ we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel.
subject: Articles
  topic: section
subject: Reconfiguration graphs
  topic: keywords
subject: $P_5$-free graphs
  topic: keywords
subject: frozen colorings
  topic: keywords
subject: $k$-mixing.
  topic: keywords
subject: 05C15
  topic: subjects
subject: 05C38
  topic: subjects
creator: Hui Lei
creator: Yulai Ma
creator: Zhengke Miao
creator: Yongtang Shi
creator: Susu Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-01-14
publication date: 2025-01-14
starting page: 394
ending page: 411
doi: 10.4208/aam.OA-2024-0024

An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media
type: item
description: We develop a locally mass-conservative enriched Petrov-Galerkin (EPG) method without any penalty term for the simulation of Darcy flow in fractured porous media. The discrete fracture model is applied to model the fractures as the lower dimensional fracture interfaces. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions in the fractures and the surrounding porous media. We propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method. The solution of the pressure can be decoupled into two steps with a standard CG method and a post-processing correction. The post-processing correction based on the bubble functions in the matrix and the fractures can be solved separately, which is useful for parallel computing. We derive a priori and a posteriori error estimates for the problem. Numerical examples are presented to illustrate the performance of the proposed method.
subject: Articles
  topic: section
subject: Discrete fracture model
  topic: keywords
subject: enriched Petrov-Galerkin method
  topic: keywords
subject: local mass conservation
  topic: keywords
subject: post-processing
  topic: keywords
subject: error analysis.
  topic: keywords
subject: 65M60
  topic: subjects
subject: 65N30
  topic: subjects
subject: 76S05
  topic: subjects
creator: Huangxin Chen
creator: Zixuan Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-01-14
publication date: 2025-01-14
starting page: 347
ending page: 393
doi: 10.4208/aam.OA-2024-0015

Existence of Global Attractor for Weakly Damped FDS Nonlinear Wave Equations
type: item
description: The paper investigates the well-posedness of global solutions and the existence of global attractors for weakly damped FDS nonlinear wave equations. It establishes the well-posedness of weak solutions using Galerkin approximation and a priori estimate. Subsequently, a dynamical system is constructed based on the well-posedness of the solution. The existence of a bounded absorbing set for the equations and the smooth properties of the operator semigroup are presented, leading to the existence of a global attractor.
subject: Articles
  topic: section
subject: Global attractors
  topic: keywords
subject: a priori estimate
  topic: keywords
subject: FDS nonlinear wave equations.
  topic: keywords
subject: 35B40
  topic: subjects
subject: 35B41
  topic: subjects
subject: 35B45
  topic: subjects
creator: Boling Guo
creator: Ying Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-01-14
publication date: 2025-01-14
starting page: 333
ending page: 346
doi: 10.4208/aam.OA-2024-0009

The Characteristics of Solutions to Semilinear Wave Equation with Logarithmic Plus Polynomial Nonlinearities
type: item
description: The semilinear wave equation with logarithmic and polynomial nonlinearities is considered in this paper. By adjusting and using potential well method, we attain the global-in-time existence and infinite time blowup solutions at subcritical initial energy level $E(0)<d.$ Then using additional conditions on initial data, these results are enlarged at critical case $E(0)=d$ and arbitrarily positive case $E(0)>0.$
subject: Articles
  topic: section
subject: Global existence
  topic: keywords
subject: blowup
  topic: keywords
subject: logarithmic and polynomial combined nonlinearity
  topic: keywords
subject: potential well.
  topic: keywords
subject: 35L05
  topic: subjects
creator: Md Salik Ahmed
creator: Weipeng Wu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-26
publication date: 2024-09-26
starting page: 262
ending page: 284
doi: 10.4208/aam.OA-2024-0011

Non-Global Existence of Regular Solution to Initial Value Problem of Relativistic Euler Equations in $\mathbb{R}^N$
type: item
description: In fluid mechanics and astrophysics, relativistic Euler equations can be used to describe the effects of special relativity which are an extension of the classical Euler equations. In this paper, we will consider the initial value problem of relativistic Euler equations in an initial bounded region of $\mathbb{R}^N.$ If the initial velocity satisfies $$\max\limits_{\vec{x_0}\in ∂Ω(0)}\sum\limits_{i=1}^N v^2_i(0,\vec{x_0})<\frac{c^2A_1}{2},$$ where $A_1$ is a positive constant depend on some sufficiently large $T^∗,$ then we can get the non-global existence of the regular solution for the $N$-dimensional relativistic Euler equations.
subject: Articles
  topic: section
subject: Non-global existence
  topic: keywords
subject: relativistic Euler equations
  topic: keywords
subject: regular solution
  topic: keywords
subject: initial value problem.
  topic: keywords
subject: 35A01
  topic: subjects
subject: 35E15
  topic: subjects
subject: 35Q75
  topic: subjects
creator: Xingli Li
creator: Jianli Liu
creator: Manwai Yuen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-26
publication date: 2024-09-26
starting page: 249
ending page: 261
doi: 10.4208/aam.OA-2024-0012

Energy Equality for the Isentropic Compressible Navier-Stokes Equations Without Upper Bound of the Density
type: item
description: In this paper, we are concerned with the minimal regularity of both the density and the velocity for the weak solutions keeping energy equality in the isentropic compressible Navier-Stokes equations. The energy equality criteria without upper bound of the density are established for the first time. Our results imply that the lower integrability of the density $\rho$ means that more integrability of the velocity $v$ or the gradient of the velocity $∇v$ are necessary for energy conservation of the isentropic compressible fluid and the inverse is also true.
subject: Articles
  topic: section
subject: Compressible Navier-Stokes equations
  topic: keywords
subject: energy equality
  topic: keywords
subject: vacuum.
  topic: keywords
subject: 35Q30
  topic: subjects
subject: 35Q35
  topic: subjects
subject: 76N06
  topic: subjects
subject: 76N10
  topic: subjects
creator: Yulin Ye
creator: Yanqing Wang
creator: Huan Yu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-26
publication date: 2024-09-26
starting page: 285
ending page: 313
doi: 10.4208/aam.OA-2024-0010

Global Classical Solution and Asymptotic Behavior to a Kind of Linearly Degenerate Quasilinear Hyperbolic System
type: item
description: This paper is concerned with the global classical solution and the asymptotic behavior to a kind of linearly degenerate quasilinear hyperbolic system in several space variables. When the semilinear terms contain at least two waves with different propagation speeds, we can prove that the system considered admits a global classical solution by the weighted energy estimate under the small and suitable decay assumptions on the initial data. Furthermore, we can show that the solution converges to a solution of the linearized system based on the decay property of the nonlinearities.
subject: Articles
  topic: section
subject: Hyperbolic system
  topic: keywords
subject: linearly degenerate
  topic: keywords
subject: positive weighted energy estimates
  topic: keywords
subject: global existence
  topic: keywords
subject: asymptotic behavior.
  topic: keywords
subject: 53C44
  topic: subjects
subject: 53C21
  topic: subjects
subject: 58J45
  topic: subjects
subject: 35L45
  topic: subjects
creator: Changhua Wei
creator: Tiantian Xing
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-26
publication date: 2024-09-26
starting page: 314
ending page: 332
doi: 10.4208/aam.OA-2024-0017

Preface: Special Issue Dedicated to Professor Avy Soffer’s 70th Birthday (Part I)
type: item
subject: Articles
  topic: section
subject:
  topic: subjects
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-26
publication date: 2024-09-26
starting page: i
ending page: iv

Entire Sign-Changing Solutions to the Fractional Critical Schrödinger Equation
type: item
description: In this paper, we consider the fractional critical Schrödinger equation (FCSE) $$(-\Delta)^su-|u|^{2^*_s-2}u=0,$$ where $u∈\dot{H}^s(\mathbb{R}^N),$ $N≥4,$ $0<s<1$ and $2^∗_s=\frac{2N}{N−2s}$ is the critical Sobolev exponent of order $s.$ By virtue of the variational method and the concentration compactness principle with the equivariant group action, we obtain some new type of nonradial, sign-changing solutions of (FCSE) in the energy space $\dot{H}^s(\mathbb{R}^N)$. The key component is that we take the equivariant group action to construct several subspace of $\dot{H}^s(\mathbb{R}^N)$ with trivial intersection, then combine the concentration compactness argument in the Sobolev space with fractional order to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of (FCSE) in $\dot{H}^s(\mathbb{R}^N).$
subject: Articles
  topic: section
subject: Fractional critical Schrödinger equation
  topic: keywords
subject: sign-changing solution
  topic: keywords
subject: the concentration-compactness principle
  topic: keywords
subject: the equivariant group action
  topic: keywords
subject: the mountain pass theorem.
  topic: keywords
subject: 35A15
  topic: subjects
subject: 35J91
  topic: subjects
subject: 35R11
  topic: subjects
creator: Xingdong Tang
creator: Guixiang Xu
creator: Chunyan Zhang
creator: Jihui Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-09-26
publication date: 2024-09-26
starting page: 219
ending page: 248
doi: 10.4208/aam.OA-2024-0006

On the Monotonicity of $Q^3$ Spectral Element Method for Laplacian
type: item
description: The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are or can be regarded as fourth order accurate finite difference schemes, which is either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of $Q^k$ spectral element methods will be lost for $k≥9$ in two dimensions, and for $k≥4$ in three dimensions. In other words, for obtaining a high order monotone scheme, only $Q^2$ and $Q^3$ spectral element methods can be unconditionally monotone in three dimensions.
subject: Articles
  topic: section
subject: Inverse positivity
  topic: keywords
subject: discrete maximum principle
  topic: keywords
subject: high order accuracy
  topic: keywords
subject: monotonicity
  topic: keywords
subject: discrete Laplacian
  topic: keywords
subject: spectral element method.
  topic: keywords
subject: 65N30
  topic: subjects
subject: 65N06
  topic: subjects
subject: 65N12
  topic: subjects
creator: Logan J. Cross
creator: Xiangxiong Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-05-11
publication date: 2024-05-11
starting page: 161
ending page: 190
doi: 10.4208/aam.OA-2024-0007

Numerical Study of Semidiscrete Penalty Approach for Stabilizing Boussinesq System with Localized Feedback Control
type: item
description: We investigate the numerical approximation for stabilizing the semidiscrete linearized Boussinesq system around an unstable stationary state. Stabilization is attained through internal feedback controls applied to the velocity and temperature equations, localized within an arbitrary open subset. This study follows the framework presented in [14], considering the continuous linearized Boussinesq system. The primary objective is to explore the penalization-based approximation of the free divergence condition in the semidiscrete case and provide a numerical validation of these results in a two-dimensional setting.
subject: Articles
  topic: section
subject: Boussinesq system
  topic: keywords
subject: penalty method
  topic: keywords
subject: stabilization
  topic: keywords
subject: feedback control.
  topic: keywords
subject: 49N10
  topic: subjects
subject: 93B05
  topic: subjects
creator: Mejdi Azaiez
creator: Kévin Le Balc’h
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-05-11
publication date: 2024-05-11
starting page: 191
ending page: 218
doi: 10.4208/aam.OA-2024-0013

On a Hybrid Method for Inverse Transmission Eigenvalue Problems
type: item
description: In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small, certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.
subject: Articles
  topic: section
subject: Inverse transmission eigenvalue problem
  topic: keywords
subject: linear sampling method
  topic: keywords
subject: neural network
  topic: keywords
subject: super-resolution.
  topic: keywords
subject: 35P25
  topic: subjects
subject: 35R30
  topic: subjects
subject: 35P15
  topic: subjects
creator: Weishi Yin
creator: Zhaobin Xu
creator: Pinchao Meng
creator: Hongyu Liu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-05-11
publication date: 2024-05-11
starting page: 139
ending page: 160
doi: 10.4208/aam.OA-2024-0003

Generalized Isospectral-Nonisospectral Modified Korteweg-de Vries Integrable Hierarchies and Related Properties
type: item
description: In this article, a new technique for deriving integrable hierarchy is discussed, i.e., such that are derived by combining the Tu scheme with the vector product. Several classes of spectral problems are introduced by three-dimensional loop algebra and six-dimensional loop algebra whose commutators are vector product, and the six-dimensional loop algebra is derived from the enlargement of the three-dimensional loop algebra. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy is worked out. The derived integrable hierarchies are reduced to the modified Korteweg-de Vries (mKdV) equation, generalized coupled mKdV integrable system and non-isospectral mKdV equation under specific parameter selection. Starting from a 3×3 matrix spectral problem, we subsequently construct an explicit $N$-fold Darboux transformation for integrable system (2.8) with the help of a gauge transformation of the corresponding spectral problem. At the same time, the determining equations of nonclassical symmetries associated with mKdV equation are presented in this paper. It follows that we investigate the coverings and the nonlocal symmetries of the nonisospectral mKdV equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.
subject: Articles
  topic: section
subject: Darboux transformation
  topic: keywords
subject: nonlocal symmetry
  topic: keywords
subject: covering
  topic: keywords
subject: manifold
  topic: keywords
subject: contact structure.
  topic: keywords
subject: 35Q51
  topic: subjects
creator: Huanhuan Lu
creator: Xinan Ren
creator: Yufeng Zhang
creator: Hongyi Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-05-11
publication date: 2024-05-11
starting page: 105
ending page: 138
doi: 10.4208/aam.OA-2024-0002

Empirical Study on Option Pricing under Markov Regime Switching Economics
type: item
description: In this research, we summarize the results of a practical study of index options based on the option valuation model which was proposed by Siu and Yang (Acta Math. Appl. Sin. Engl. Ser., 25(3) (2009), pp. 339–388), where an EMM kernel is integrated which takes into account all risk components of a regime-switching model. Further, the regime-switching risk of an economy in the options is priced using a hidden Markov regime-switching model with the risky underlying asset being modulated by a discrete-time, finite-state, hidden Markov chain whose states represent the hidden states of an economy. We apply such a model to the pricing of Hang Seng Index options based on the real-world financial data from October 2009 to October 2010 (i.e., for the year in which the model was proposed). We employed the entropy martingale measure (EMM) approach proposed by Siu and Yang (Acta Math. Appl. Sin. Engl. Ser., 25(3) (2009), pp. 339–388) to determine the optimal martingale measure for the Markov-modulated GBM. In addition, we have proposed a numerical technique called the weighted difference method to compliment the EMM approach. We have also verified the extended jump-diffusion model under regime-switching that we proposed recently (Int. J. Finan. Eng., 6(4) (2019), 1950038) using the 50ETF options which are obtained from Shanghai Stock Exchange covering a time span from January 2018 to December 2022. Further, we have highlighted the challenges for the EMM kernel-based Markov regime-switching model for pricing the out-of-the-money index options in the real world.
subject: Articles
  topic: section
subject: Option pricing
  topic: keywords
subject: EMM
  topic: keywords
subject: regime-switching
  topic: keywords
subject: hidden Markov model
  topic: keywords
subject: Esscher transform
  topic: keywords
subject: weighted difference method.
  topic: keywords
subject: 91G20
  topic: subjects
subject: 91G60
  topic: subjects
creator: Lianfeng (David) Liu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-02-29
publication date: 2024-02-29
starting page: 21
ending page: 42
doi: 10.4208/aam.OA-2023-0012

Error Analysis of the Nonconforming $P_1$ Finite Element Method to the Sequential Regularization Formulation for Unsteady Navier-Stokes Equations
type: item
description: In this paper we investigate the nonconforming $P_1$ finite element approximation to the sequential regularization method for unsteady Navier-Stokes equations. We provide error estimates for a full discretization scheme. Typically, conforming $P_1$ finite element methods lead to error bounds that depend inversely on the penalty parameter $\epsilon.$ We obtain an $\epsilon$-uniform error bound by utilizing the nonconforming $P_1$ finite element method in this paper. Numerical examples are given to verify theoretical results.
subject: Articles
  topic: section
subject: Navier-Stokes equations
  topic: keywords
subject: error estimates
  topic: keywords
subject: finite element method
  topic: keywords
subject: stabilization method.
  topic: keywords
subject: 65M60
  topic: subjects
subject: 76D05
  topic: subjects
creator: Yanming Lai
creator: Kewei Liang
creator: Ping Lin
creator: Xiliang Lu
creator: Qimeng Quan
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2024-02-29
publication date: 2024-02-29
starting page: 43
ending page: 70
doi: 10.4208/aam.OA-2023-0016