Methods and applications of analysis最新文献

{"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":" ","pages":""},"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}

{"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":" ","pages":""},"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}

{"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":"1 1","pages":""},"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}

{"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":"1 1","pages":""},"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}

{"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":"1 1","pages":""},"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}

{"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":"1 1","pages":""},"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}

{"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":"8 1","pages":""},"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}

{"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":"1 1","pages":""},"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}

{"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":"1 1","pages":""},"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}

{"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":"1 1","pages":""},"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}