Pub Date : 2019-07-03DOI: 10.4310/MAA.2020.v27.n2.a4
Yuan Gao, Jian‐Guo Liu
In this paper we study the relaxation process of Peierls-Nabarro dislocation model, which is a gradient flow with singular nonlocal energy and double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. We prove the dynamic solution to Peierls-Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.
{"title":"Long time behavior of dynamic solution to Peierls–Nabarro dislocation model","authors":"Yuan Gao, Jian‐Guo Liu","doi":"10.4310/MAA.2020.v27.n2.a4","DOIUrl":"https://doi.org/10.4310/MAA.2020.v27.n2.a4","url":null,"abstract":"In this paper we study the relaxation process of Peierls-Nabarro dislocation model, which is a gradient flow with singular nonlocal energy and double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. We prove the dynamic solution to Peierls-Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48255880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-28DOI: 10.4310/MAA.2020.v27.n4.a2
Paul Bryan, Heiko Kroner, Julian Scheuer
We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0
我们用截面曲率为正下界的Einstein流形中平均曲率的幂$p$，$0
{"title":"Li–Yau gradient estimates for curvature flows in positively curved manifolds","authors":"Paul Bryan, Heiko Kroner, Julian Scheuer","doi":"10.4310/MAA.2020.v27.n4.a2","DOIUrl":"https://doi.org/10.4310/MAA.2020.v27.n4.a2","url":null,"abstract":"We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower bound is sufficiently large compared to the derivatives of the curvature tensor of the ambient space and that the mean curvature of the initial hypersurface is sufficiently large compared to the ambient geometry. We also obtain some new Harnack inequalities for more general curvature flows in the sphere, as well as a monotonicity estimate for the mean curvature flow in non-negatively curved, locally symmetric spaces.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47957229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n2.a6
É. Gauci, A. Belme, A. Carabias, A. Loseille, F. Alauzet, A. Dervieux
{"title":"A priori error-based mesh adaptation in CFD","authors":"É. Gauci, A. Belme, A. Carabias, A. Loseille, F. Alauzet, A. Dervieux","doi":"10.4310/maa.2019.v26.n2.a6","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n2.a6","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n1.a1
Xue Luo, Fang Wang
{"title":"On the Newton polyhedrons with one inner lattice point","authors":"Xue Luo, Fang Wang","doi":"10.4310/maa.2019.v26.n1.a1","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n1.a1","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n2.a1
G. Bastin, J. Coron
We consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral (PI) boundary control. For channels with uniform steady states, the analysis has been carried out previously in the literature with spectral methods as well as with Lyapunov functions in Riemann coordinates. In this article, our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapunov function in physical coordinates. Introduction The hyperbolic Saint-Venant equations are commonly used for the description of water flow dynamics in open channels and for the design of management and control systems in irrigation networks and navigable rivers. In particular, the exponential stabilization of Saint-Venant equations by boundary feedback control has been a recurring research topic in the literature for more than twenty years. The earlier results dealt with static proportional control. In the simplest case of horizontal channels with negligible friction, the stability analysis was carried out in [6] with an entropy Lyapunov function, in [16, 11] with the method of characteristics, and in [7, Section VI] with a Lyapunov function in Riemann coordinates. The stability analysis was then extended to channels with slope and friction. In the special case of a uniform steady state, the stability analysis was carried out with a spectral method for linearized equations in [17, Section 6]. However the linearized system stability does not directly imply the stability of the steady state for the nonlinear SaintVenant equations (see e.g. [8]). For this nonlinear case, the stability analysis is done in [4, 13] with a Lyapunov function in Riemann coordinates. More recently, the case of channels with friction and slope and non-uniform steady state was considered in [3] and [15] with dedicated Lyapunov functions expressed in physical coordinates. The boundary feedback stabilization of Saint-Venant equations by Proportional-Integral (PI) control has received much less attention in the literature. It has been analyzed for channels with uniform steady states in [5] with a spectral method and in [14, Section 4], [2, Section 5.5] with Lyapunov functions in Riemann coordinates. In the present article, our main contribution is to show how the analysis of [3] can be extended to channels with non-uniform steady states under PI control, using a Lyapunov function in physical coordinates. Obviously, in principle, stabilization is also possible with more sophisticated control laws. In particular, the recent backstepping method for 2 × 2 hyperbolic systems, see e.g. [10, 1, 12], ∗Department of Mathematical Engineering, ICTEAM, University of Louvain, Louvain-La-Neuve, Belgium. †Sorbonne Université, Université Paris-Diderot SPC, CNRS,
我们考虑由Saint-Venant方程表示的开放通道，这些通道在下游边界受到监测和控制，并在上游边界受到不可测量的流动干扰。我们讨论了比例积分(PI)边界控制下的反馈镇定和干扰抑制问题。对于具有均匀稳态的通道，先前的文献中已经使用谱方法以及黎曼坐标中的李雅普诺夫函数进行了分析。在本文中，我们的主要贡献是展示如何将分析扩展到具有物理坐标中的李雅普诺夫函数的非均匀稳态的通道。双曲圣维南方程通常用于描述明渠中的水流动力学，以及用于灌溉网络和通航河流的管理和控制系统的设计。特别是，边界反馈控制的Saint-Venant方程的指数镇定问题已经成为二十多年来文献中反复出现的研究课题。先前的结果处理静态比例控制。对于最简单的可忽略摩擦的水平通道，在[6]中使用熵Lyapunov函数进行稳定性分析，在[16,11]中使用特征方法进行稳定性分析，在[7,Section VI]中使用黎曼坐标中的Lyapunov函数进行稳定性分析。然后将稳定性分析扩展到有坡度和摩擦力的河道。在均匀稳态的特殊情况下，用谱法对[17，第6节]中的线性化方程进行稳定性分析。然而，线性化的系统稳定性并不直接意味着非线性SaintVenant方程的稳态稳定性(例如[8])。对于这种非线性情况，稳定性分析在[4,13]中使用黎曼坐标下的Lyapunov函数进行。最近，在[3]和[15]中考虑了具有摩擦和斜率和非均匀稳态的通道的情况，并使用物理坐标表示的专用Lyapunov函数。基于比例积分控制的Saint-Venant方程边界反馈镇定问题在文献中得到的关注较少。用谱法分析了[5]中均匀稳态的信道，并在[14，第4节]、[2，第5.5节]中用黎曼坐标中的Lyapunov函数分析了信道。在本文中，我们的主要贡献是展示如何使用物理坐标中的李雅普诺夫函数将[3]的分析扩展到PI控制下具有非均匀稳态的通道。显然，原则上，更复杂的控制律也可以实现稳定。特别地，最近的关于2 × 2双曲系统的反演方法，见e.g.[10,1,12]，∗，University of Louvain- la - neuve, University of Louvain。†索邦大学，巴黎狄德罗大学SPC, CNRS, INRIA, Jacques-Louis Lions实验室，LJLL, quipe CAGE, F-75005巴黎，法国。
{"title":"Exponential stability of PI control for Saint-Venant equations with a friction term","authors":"G. Bastin, J. Coron","doi":"10.4310/maa.2019.v26.n2.a1","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n2.a1","url":null,"abstract":"We consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral (PI) boundary control. For channels with uniform steady states, the analysis has been carried out previously in the literature with spectral methods as well as with Lyapunov functions in Riemann coordinates. In this article, our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapunov function in physical coordinates. Introduction The hyperbolic Saint-Venant equations are commonly used for the description of water flow dynamics in open channels and for the design of management and control systems in irrigation networks and navigable rivers. In particular, the exponential stabilization of Saint-Venant equations by boundary feedback control has been a recurring research topic in the literature for more than twenty years. The earlier results dealt with static proportional control. In the simplest case of horizontal channels with negligible friction, the stability analysis was carried out in [6] with an entropy Lyapunov function, in [16, 11] with the method of characteristics, and in [7, Section VI] with a Lyapunov function in Riemann coordinates. The stability analysis was then extended to channels with slope and friction. In the special case of a uniform steady state, the stability analysis was carried out with a spectral method for linearized equations in [17, Section 6]. However the linearized system stability does not directly imply the stability of the steady state for the nonlinear SaintVenant equations (see e.g. [8]). For this nonlinear case, the stability analysis is done in [4, 13] with a Lyapunov function in Riemann coordinates. More recently, the case of channels with friction and slope and non-uniform steady state was considered in [3] and [15] with dedicated Lyapunov functions expressed in physical coordinates. The boundary feedback stabilization of Saint-Venant equations by Proportional-Integral (PI) control has received much less attention in the literature. It has been analyzed for channels with uniform steady states in [5] with a spectral method and in [14, Section 4], [2, Section 5.5] with Lyapunov functions in Riemann coordinates. In the present article, our main contribution is to show how the analysis of [3] can be extended to channels with non-uniform steady states under PI control, using a Lyapunov function in physical coordinates. Obviously, in principle, stabilization is also possible with more sophisticated control laws. In particular, the recent backstepping method for 2 × 2 hyperbolic systems, see e.g. [10, 1, 12], ∗Department of Mathematical Engineering, ICTEAM, University of Louvain, Louvain-La-Neuve, Belgium. †Sorbonne Université, Université Paris-Diderot SPC, CNRS,","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n2.a2
A. Caboussat
The numerical solution of the Dirichlet problem for an elliptic Pucci’s equation in two dimensions of space is addressed by using a least-squares approach. The algorithm relies on an iterative relaxation method that decouples a variational linear elliptic PDE problem from the local nonlinearities. The approximation method relies on mixed low order finite element methods. The least-squares framework allows to revisit and extend the approach and the results presented in (Caffarelli, Glowinski, 2008) to more general cases. Numerical results show the convergence of the iterative sequence to the exact solution, when such a solution exists. The robustness of the approach is highlighted, when dealing with various types of meshes, domains with curved boundaries, nonconvex domains, or non-smooth solutions.
{"title":"Least-squares/relaxation method for the numerical solution of a 2D Pucci’s equation","authors":"A. Caboussat","doi":"10.4310/maa.2019.v26.n2.a2","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n2.a2","url":null,"abstract":"The numerical solution of the Dirichlet problem for an elliptic Pucci’s equation in two dimensions of space is addressed by using a least-squares approach. The algorithm relies on an iterative relaxation method that decouples a variational linear elliptic PDE problem from the local nonlinearities. The approximation method relies on mixed low order finite element methods. The least-squares framework allows to revisit and extend the approach and the results presented in (Caffarelli, Glowinski, 2008) to more general cases. Numerical results show the convergence of the iterative sequence to the exact solution, when such a solution exists. The robustness of the approach is highlighted, when dealing with various types of meshes, domains with curved boundaries, nonconvex domains, or non-smooth solutions.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n4.a1
Nan Jiang
{"title":"On the wavewise entropy inequalities for high-resolution schemes with source terms II: the fully-discrete case","authors":"Nan Jiang","doi":"10.4310/maa.2019.v26.n4.a1","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n4.a1","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n3.a1
T. Kärkkäinen, R. Glowinski
{"title":"A Douglas–Rachford method for sparse extreme learning machine","authors":"T. Kärkkäinen, R. Glowinski","doi":"10.4310/maa.2019.v26.n3.a1","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n3.a1","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n3.a4
Haixia Liu, X. Tai
{"title":"Curvature-based authentication of van Gogh paintings","authors":"Haixia Liu, X. Tai","doi":"10.4310/maa.2019.v26.n3.a4","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n3.a4","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n4.a3
Fares Bensaid, M. Moussai
{"title":"Realizations of the homogeneous Besov-type spaces","authors":"Fares Bensaid, M. Moussai","doi":"10.4310/maa.2019.v26.n4.a3","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n4.a3","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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