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Unified asymptotic analysis and numerical simulations of singularly perturbed linear differential equations under various nonlocal boundary effects 各种非局部边界效应下奇异扰动线性微分方程的统一渐近分析和数值模拟
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-02-01 DOI: 10.4310/cms.2024.v22.n2.a5
Xianjin Chen, Chiun-Chang Lee, Masashi Mizuno
{"title":"Unified asymptotic analysis and numerical simulations of singularly perturbed linear differential equations under various nonlocal boundary effects","authors":"Xianjin Chen, Chiun-Chang Lee, Masashi Mizuno","doi":"10.4310/cms.2024.v22.n2.a5","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a5","url":null,"abstract":"While being concerned with a singularly perturbed linear differential equation subject to integral boundary conditions, the exact solutions, in general, cannot be specified, and the validity of the maximum principle is unassurable. Hence, a problem arises: <i>how to identify the boundary asymptotics more precisely?</i> We develop a rigorous asymptotic method involving recovered boundary data to tackle the problem. A key ingredient of the approach is to transform the “nonlocal” boundary conditions into “local” boundary conditions. Then, we perform an “$varepsilon log varepsilon$-estimate” to obtain the refined boundary asymptotics of its solutions with respect to the singular perturbation parameter $varepsilon$. Furthermore, for the inhomogeneous case, diversified asymptotic behaviors including uniform boundedness and asymptotic blow-up are obtained. Numerical simulations and validations are also presented to further support the corresponding theoretical results.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lifespan estimates of solutions to the weakly coupled system of semilinear wave equations with space dependent dampings 具有空间相关阻尼的半线性波方程弱耦合系统解的寿命估计
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-02-01 DOI: 10.4310/cms.2024.v22.n2.a4
Sen Ming, Han Yang, Xiongmei Fan
{"title":"Lifespan estimates of solutions to the weakly coupled system of semilinear wave equations with space dependent dampings","authors":"Sen Ming, Han Yang, Xiongmei Fan","doi":"10.4310/cms.2024.v22.n2.a4","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a4","url":null,"abstract":"$def lv{lvert}defrv{rvert}$ This paper is devoted to investigating the weakly coupled system of semilinear wave equations with space dependent dampings and power nonlinearities ${lv v rv}^p, {lv u rv}^q$, derivative nonlinearities ${lv v_t rv}^p, {lv u_t rv}^q$, mixed nonlinearities ${lv v rv}^q, {lv u_t rv}^p$, combined nonlinearities ${lv v_t rv}^{p_1} + {lv v rv}^{q_1}, {lv u_t rv}^{p_2} + {lv u rv}^{q_2}$, combined and power nonlinearities ${lv v_t rv}^{p_1} + {lv v rv}^{q_1}, {lv u rv}^{q_2}$, combined and derivative nonlinearities ${lv v_t rv}^{p_1} + {lv v rv}^{q_1}, {lv u_t rv}^{p_2}$, respectively. Formation of singularities and lifespan estimates of solutions to the problem in the sub-critical and critical cases are illustrated by making use of test function technique. The main innovation is that upper bound lifespan estimates of solutions are associated with the Strauss exponent and Glassey exponent.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations 粘性接触波对一维可压缩纳维-斯托克斯方程的稳定性和衰减率
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-02-01 DOI: 10.4310/cms.2024.v22.n2.a2
Xinxiang Bian, Lingling Xie
{"title":"Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations","authors":"Xinxiang Bian, Lingling Xie","doi":"10.4310/cms.2024.v22.n2.a2","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a2","url":null,"abstract":"This paper studies the large-time asymptotic stability and optimal time-decay rate of viscous contact wave to one-dimensional compressible Navier–Stokes equations. We prove that one-dimensional compressible Navier–Stokes equations are asymptotically stable for viscous contact wave with arbitrarily large strength, under large initial perturbations. The time optimal decay rate of viscous contact wave is also obtained under the small initial perturbations. In the proof, the Lagrange transform is used to cancel the convection terms, which are difficult to estimate due to the lower spatial derivatives compared with the diffusion terms.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global existence of perturbed Navier–Stokes system around Landau solutions with slowly decaying oscillation 具有缓慢衰减振荡的朗道解周围扰动纳维-斯托克斯系统的全局存在性
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-02-01 DOI: 10.4310/cms.2024.v22.n2.a10
Jiayan Wu, Cuili Zhai, Jingjing Zhang, Ting Zhang
{"title":"Global existence of perturbed Navier–Stokes system around Landau solutions with slowly decaying oscillation","authors":"Jiayan Wu, Cuili Zhai, Jingjing Zhang, Ting Zhang","doi":"10.4310/cms.2024.v22.n2.a10","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a10","url":null,"abstract":"In this paper, we consider the perturbed Navier–Stokes system around the Landau solutions. Using the energy method and the continuation method, we show the global existence of the $L^2$ local energy solution for the perturbed Navier–Stokes system with the oscillation decay initial data $v_0 in E^2_{sigma} + L^3_{operatorname{uloc}} ,$.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A stochastic Galerkin method for the direct and inverse random source problems of the Helmholtz equation 亥姆霍兹方程直接和反向随机源问题的随机伽勒金方法
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2024-02-01 DOI: 10.4310/cms.2024.v22.n2.a11
Ning Guan, Dingyu Chen, Peijun Li, Xinghui Zhong
{"title":"A stochastic Galerkin method for the direct and inverse random source problems of the Helmholtz equation","authors":"Ning Guan, Dingyu Chen, Peijun Li, Xinghui Zhong","doi":"10.4310/cms.2024.v22.n2.a11","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a11","url":null,"abstract":"This paper investigates a novel approach for solving both the direct and inverse random source problems of the one-dimensional Helmholtz equation with additive white noise, based on the generalized polynomial chaos (gPC) approximation. The direct problem is to determine the wave field that is emitted from a random source, while the inverse problem is to use the boundary measurements of the wave field at various frequencies to reconstruct the mean and variance of the source. The stochastic Helmholtz equation is reformulated in such a way that the random source is represented by a collection of mutually independent random variables. The stochastic Galerkin method is employed to transform the model equation into a two-point boundary value problem for the gPC expansion coefficients. The explicit connection between the sine or cosine transform of the mean and variance of the random source and the analytical solutions for the gPC coefficients is established. The advantage of these analytical solutions is that the gPC coefficients are zero for basis polynomials of degree higher than one, which implies that the total number of the gPC basis functions increases proportionally to the dimension, and indicates that the stochastic Galerkin method has the potential to be used in practical applications involving random variables of higher dimensions. By taking the inverse sine or cosine transform of the data, the inverse problem can be solved, and the statistical information of the random source such as the mean and variance can be obtained straightforwardly as the gPC basis functions are orthogonal. Numerical experiments are conducted to demonstrate the efficiency of the proposed method.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Description of random level sets by polynomial chaos expansions 用多项式混沌展开描述随机水平集
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2023-12-07 DOI: 10.4310/cms.2024.v22.n1.a4
Markus Bambach, Stephan Gerster, Michael Herty, Aleksey Sikstel
{"title":"Description of random level sets by polynomial chaos expansions","authors":"Markus Bambach, Stephan Gerster, Michael Herty, Aleksey Sikstel","doi":"10.4310/cms.2024.v22.n1.a4","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a4","url":null,"abstract":"We present a novel approach to determine the evolution of level sets under uncertainties in their velocity fields. This leads to a stochastic description of level sets. To compute the quantiles of random level sets, we use the stochastic Galerkin method for a hyperbolic reformulation of the equations for the propagation of level sets. A novel intrusive Galerkin formulation is presented and proven to be hyperbolic. It induces a corresponding finite-volume scheme that is specifically tailored to uncertain velocities.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A new priori error estimation of nonconforming element for two-dimensional linearly elastic shallow shell equations 二维线性弹性浅壳方程中不符元素的新先验误差估计
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2023-12-07 DOI: 10.4310/cms.2024.v22.n1.a7
Rongfang Wu, Xiaoqin Shen, Qian Yang, Shengfeng Zhu
{"title":"A new priori error estimation of nonconforming element for two-dimensional linearly elastic shallow shell equations","authors":"Rongfang Wu, Xiaoqin Shen, Qian Yang, Shengfeng Zhu","doi":"10.4310/cms.2024.v22.n1.a7","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a7","url":null,"abstract":"In this paper, we mainly propose a new <i>priori</i> error estimation for the two-dimensional linearly elastic shallow shell equations, which rely on a family of Kirchhoff–Love theories. As the displacement components of the points on the middle surface have different regularities, the nonconforming element for the discretization shallow shell equations is analysed. Then, relying on the enriching operator, a new error estimate of energy norm is given under the regularity assumption $vec{zeta}_H times zeta_3 in (H^{1+m} (omega))^2 times H^{2+m} (omega)$ with any $m gt 0$. Compared with the classic error analysis in other shell literature, convergence order of numerical solution can be controlled by its corresponding approximation error with an arbitrarily high order term, which fills the gap in the computational shell theory. Finally, numerical results for the saddle shell and cylindrical shell confirm the theoretical prediction.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unipolar Euler–Poisson equations with time-dependent damping: blow-up and global existence 具有随时间变化的阻尼的单极欧拉-泊松方程：爆炸和全局存在性
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2023-12-07 DOI: 10.4310/cms.2024.v22.n1.a8
Jianing Xu, Shaohua Chen, Ming Mei, Yuming Qin
{"title":"Unipolar Euler–Poisson equations with time-dependent damping: blow-up and global existence","authors":"Jianing Xu, Shaohua Chen, Ming Mei, Yuming Qin","doi":"10.4310/cms.2024.v22.n1.a8","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a8","url":null,"abstract":"This paper is concerned with the Cauchy problem for one-dimensional unipolar Euler–Poisson equations with time-dependent damping, where the time-asymptotically degenerate damping in the form of $-dfrac{mu}{(1+t)^lambda} rho mu$ for $lambda gt 0$ with $mu gt 0$ plays a crucial role for the structure of solutions. The main issue of the paper is to investigate the critical case with $lambda=1$. We first prove that, for all cases with $lambda gt 0$ and $mu gt 0$ (including the critical case of $lambda=1$), once the initial data is steep at a point, then the solutions are locally bounded but their derivatives will blow up in finite time, by means of the method of Riemann invariants and the technical convex analysis. Secondly, for the critical case of $lambda=1$ with $mu gt 7/3$, we prove that there exists a unique global solution, once the initial perturbation around the constant steady-state is sufficiently small. In particular, we derive the algebraic convergence rates of the solution to the constant steady-state, which are piecewise, related to the parameter $mu$ for $7/3 lt mu leq 3$, $3 lt mu leq 4$ and $mu gt 4$. The adopted method of proof in this critical case is the technical time-weighted energy method and the time-weight depends on the parameter $mu$. Finally, we carry out some numerical simulations in two cases for blow-up and global existence, respectively, which numerically confirm our theoretical results.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A general framework for nonlocal Neumann problems 非局部 Neumann 问题的一般框架
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2023-12-07 DOI: 10.4310/cms.2024.v22.n1.a2
Guy Foghem, Moritz Kassmann
{"title":"A general framework for nonlocal Neumann problems","authors":"Guy Foghem, Moritz Kassmann","doi":"10.4310/cms.2024.v22.n1.a2","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a2","url":null,"abstract":"Within the framework of Hilbert spaces, we solve nonlocal problems in bounded domains with prescribed conditions on the complement of the domain. Our main focus is on the inhomogeneous Neumann problem in a rather general setting. We also study the transition from exterior value problems to local boundary value problems. Several results are new even for the fractional Laplace operator. The setting also covers relevant models in the framework of peridynamics.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 20
Energy method for the Boltzmann equation of monatomic gaseous mixtures 单原子气体混合物波尔兹曼方程的能量法
IF 1 4区 数学
Communications in Mathematical Sciences Pub Date : 2023-12-07 DOI: 10.4310/cms.2024.v22.n1.a6
Laurent Boudin, Bérénice Grec, Milana Pavić-Čolić, Srboljub Simić
{"title":"Energy method for the Boltzmann equation of monatomic gaseous mixtures","authors":"Laurent Boudin, Bérénice Grec, Milana Pavić-Čolić, Srboljub Simić","doi":"10.4310/cms.2024.v22.n1.a6","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a6","url":null,"abstract":"In this paper, we present an energy method for the system of Boltzmann equations in the multicomponent mixture case, based on a micro-macro decomposition. More precisely, the perturbation of a solution to the Boltzmann equation around a global equilibrium is decomposed into the sum of a macroscopic and a microscopic part, for which we obtain a priori estimates at both lower and higher orders. These estimates are obtained under a suitable smallness assumption. The assumption can be justified a posteriori in the higher-order case, leading to the closure of the corresponding estimate.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
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