Pub Date : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a5
Xianjin Chen, Chiun-Chang Lee, Masashi Mizuno
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: how to identify the boundary asymptotics more precisely? 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.
{"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":"37 1","pages":""},"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}
Pub Date : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a4
Sen Ming, Han Yang, Xiongmei Fan
$def lv{lvert}defrv{rvert}$ 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.
$def lv{lvert}defrv{rvert}$ 本文致力于研究半线性波方程的弱耦合系统，该系统具有空间相关阻尼和功率非线性特性 ${lv v rv}^p、{导数非线性 ${lv v_t rv}^p,{lv u_t rv}^q$,混合非线性 ${lv v rv}^q,{lv u_t rv}^p$,组合非线性 ${lv v_t rv}^{p_1}。+ {lv v rv}^{q_1}, {lv u_t rv}^{p_2}+ {lv u rv}^{q_2}$，组合非线性和功率非线性 ${lv v_t rv}^{p_1}。+ {lv v rv}^{q_1}，{lv u rv}^{q_2}$，组合和导数非线性 ${lv v_t rv}^{p_1}。+ {lv v rv}^{q_1}，{lv u_t rv}^{p_2}$。利用检验函数技术说明了在次临界和临界情况下问题解的奇点形成和寿命估计。主要创新之处在于解的上限寿命估计值与 Strauss 指数和 Glassey 指数相关联。
{"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":"15 1","pages":""},"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}
Pub Date : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a2
Xinxiang Bian, Lingling Xie
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.
{"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":"71 1","pages":""},"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}
Pub Date : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a10
Jiayan Wu, Cuili Zhai, Jingjing Zhang, Ting Zhang
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 $v_0 in E^2_{sigma} + L^3_{operatorname{uloc}} ,$.
本文考虑了围绕朗道解的扰动纳维-斯托克斯系统。利用能量法和延续法，我们证明了具有振荡衰减初始数据 $v_0 in E^2_{sigma} + L^3_{operatorname{uloc}} 的扰动纳维-斯托克斯系统的 $L^2$ 局域能量解的全局存在性。,$.
{"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":"122 1","pages":""},"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}
Pub Date : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a11
Ning Guan, Dingyu Chen, Peijun Li, Xinghui Zhong
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.
{"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":"37 1","pages":""},"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}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a4
Markus Bambach, Stephan Gerster, Michael Herty, Aleksey Sikstel
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.
{"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":"149 1","pages":""},"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}
In this paper, we mainly propose a new priori 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.
本文主要针对二维线性弹性浅壳方程提出了一种新的先验误差估计方法，该方法依赖于基尔霍夫-洛夫理论族。由于中面上各点的位移分量具有不同的规律性，因此分析了离散化浅壳方程的非符合元素。然后，依靠富集算子，在任意 $m gt 0$ 的正则假设 $vec{zeta}_H times zeta_3 in (H^{1+m} (omega))^2 times H^{2+m} (omega)$ 下给出了能量规范的新误差估计。与其他壳文献中的经典误差分析相比，数值解的收敛阶数可以由其对应的近似误差以任意高阶项来控制，这填补了计算壳理论的空白。最后，鞍形壳和圆柱形壳的数值结果证实了理论预测。
{"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":"23 1","pages":""},"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}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a8
Jianing Xu, Shaohua Chen, Ming Mei, Yuming Qin
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.
{"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":"29 1","pages":""},"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}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a2
Guy Foghem, Moritz Kassmann
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.
{"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":"29 1","pages":""},"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}
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.
{"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":"84 1","pages":""},"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}
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