Pub Date : 2018-01-01DOI: 10.4310/MAA.2018.V25.N1.A1
H. Freistühler
. Starting from a variational interpretation of enthalpy, this paper formulates a rela- tivistically covariant version of the Euler-Korteweg equations of fluid dynamics. The system has a canonical Lagrangian and converges in the non-relativistic limit to Dunn and Serrin’s formulation.
{"title":"A relativistic version of the Euler–Korteweg equations","authors":"H. Freistühler","doi":"10.4310/MAA.2018.V25.N1.A1","DOIUrl":"https://doi.org/10.4310/MAA.2018.V25.N1.A1","url":null,"abstract":". Starting from a variational interpretation of enthalpy, this paper formulates a rela- tivistically covariant version of the Euler-Korteweg equations of fluid dynamics. The system has a canonical Lagrangian and converges in the non-relativistic limit to Dunn and Serrin’s formulation.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488455","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 : 2018-01-01DOI: 10.4310/maa.2018.v25.n4.a3
Naveed Hussain
. Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : ( C n , 0) → ( C , 0). Let L ( V ) be the Lie algebra of derivations of the moduli algebra A ( V ) := O n / ( f,∂f/∂x 1 , ··· ,∂f/∂x n ), i.e., L ( V ) = Der( A ( V ) ,A ( V )). The Lie algebra L ( V ) is finite dimensional solvable algebra and plays an important role in singularity theory. According to Elashvili and Khimshiashvili ([15], [23]) L ( V ) is called Yau algebra and the dimension of L ( V ) is called Yau number. The studies of finite dimensional Lie algebras L ( V ) that arising from isolated singularities was started by Yau [44] and has been systematically studied by Yau, Zuo and their coauthors. Most studies of Lie algebras L ( V ) were oriented to classify the isolated singularities. This work surveys the researches on Yau algebras L ( V ) of isolated singularities.
． 设V是一个原点有孤立奇点的超曲面，其定义为全纯函数f: (cn, 0)→(c0, 0)。设L (V)是模代数a (V)的派生李代数:= O n / (f，∂f/∂x 1，···，∂f/∂x n)，即L (V) = Der(a (V)， a (V))。李代数L (V)是有限维可解代数，在奇点理论中起着重要的作用。根据Elashvili和Khimshiashvili([15]，[23])的说法，L (V)称为Yau代数，L (V)的维数称为Yau数。由孤立奇点产生的有限维李代数L (V)的研究是由Yau b[44]开始的，并由Yau、Zuo和他们的合作者进行了系统的研究。李代数L (V)的研究大多是针对孤立奇点的分类。本文综述了孤立奇点的Yau代数L (V)的研究。
{"title":"Survey on derivation Lie algebras of isolated singularities","authors":"Naveed Hussain","doi":"10.4310/maa.2018.v25.n4.a3","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n4.a3","url":null,"abstract":". Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : ( C n , 0) → ( C , 0). Let L ( V ) be the Lie algebra of derivations of the moduli algebra A ( V ) := O n / ( f,∂f/∂x 1 , ··· ,∂f/∂x n ), i.e., L ( V ) = Der( A ( V ) ,A ( V )). The Lie algebra L ( V ) is finite dimensional solvable algebra and plays an important role in singularity theory. According to Elashvili and Khimshiashvili ([15], [23]) L ( V ) is called Yau algebra and the dimension of L ( V ) is called Yau number. The studies of finite dimensional Lie algebras L ( V ) that arising from isolated singularities was started by Yau [44] and has been systematically studied by Yau, Zuo and their coauthors. Most studies of Lie algebras L ( V ) were oriented to classify the isolated singularities. This work surveys the researches on Yau algebras L ( V ) of isolated singularities.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489021","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 : 2018-01-01DOI: 10.4310/maa.2018.v25.n3.a2
Chong-qing Cheng, Min Zhou
{"title":"Introduction to the study of Arnold diffusion","authors":"Chong-qing Cheng, Min Zhou","doi":"10.4310/maa.2018.v25.n3.a2","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n3.a2","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488882","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 : 2018-01-01DOI: 10.4310/MAA.2018.V25.N2.A1
K. Yamazaki
. In regards to the mathematical issue of whether a system of equations admits a unique solution for all time or not, given an arbitrary initial data sufficiently smooth, the case of the magnetohydrodynamics system may be arguably more difficult than that of the Navier-Stokes equations. In the last several years, an explosive amount of work by many mathematicians was devoted to make progress toward the global well-posedness of the two-dimensional magnetohydro- dynamics system with diffusion in terms of a full Laplacian but with zero dissipation; nevertheless, this problem remains open. The purpose of this manuscript is to provide a second proof of the global well-posedness in case the diffusion is in the form of a full Laplacian, and the dissipation is in the form of a fractional Laplacian with an arbitrary small power. In contrast to the first proof of this result in the literature that took advantage of the property of a heat kernel, the main tools in this manuscript consist of Besov space techniques, in particular fractional chain rule, which has been proven to possess potentials to lead to resolutions of difficult problems, in particular of fluid dynamics partial differential equations.
{"title":"Second proof of the global regularity of the two-dimensional MHD system with full diffusion and arbitrary weak dissipation","authors":"K. Yamazaki","doi":"10.4310/MAA.2018.V25.N2.A1","DOIUrl":"https://doi.org/10.4310/MAA.2018.V25.N2.A1","url":null,"abstract":". In regards to the mathematical issue of whether a system of equations admits a unique solution for all time or not, given an arbitrary initial data sufficiently smooth, the case of the magnetohydrodynamics system may be arguably more difficult than that of the Navier-Stokes equations. In the last several years, an explosive amount of work by many mathematicians was devoted to make progress toward the global well-posedness of the two-dimensional magnetohydro- dynamics system with diffusion in terms of a full Laplacian but with zero dissipation; nevertheless, this problem remains open. The purpose of this manuscript is to provide a second proof of the global well-posedness in case the diffusion is in the form of a full Laplacian, and the dissipation is in the form of a fractional Laplacian with an arbitrary small power. In contrast to the first proof of this result in the literature that took advantage of the property of a heat kernel, the main tools in this manuscript consist of Besov space techniques, in particular fractional chain rule, which has been proven to possess potentials to lead to resolutions of difficult problems, in particular of fluid dynamics partial differential equations.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488993","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 : 2018-01-01DOI: 10.4310/maa.2018.v25.n4.a2
Seng Hu
{"title":"On John Mather’s work","authors":"Seng Hu","doi":"10.4310/maa.2018.v25.n4.a2","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n4.a2","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489013","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 : 2018-01-01DOI: 10.4310/MAA.2018.V25.N2.A4
Taynara B. De Souza, W. Melo, P. Zingano
. In this paper, the authors establish lower bounds for the usual Lebesgue norms of the maximal solution of the Magnetohydrodynamics Equations and present some criteria for global existence of solution. Thus, we can understand better on the blow-up behavior of this same solution. In addition, it is important to point out that we reach our main results by using standard techniques obtained from Navier-Stokes Equations.
{"title":"On lower bounds for the solution, and its spatial derivatives, of the Magnetohydrodynamics Equations in Lebesgue spaces","authors":"Taynara B. De Souza, W. Melo, P. Zingano","doi":"10.4310/MAA.2018.V25.N2.A4","DOIUrl":"https://doi.org/10.4310/MAA.2018.V25.N2.A4","url":null,"abstract":". In this paper, the authors establish lower bounds for the usual Lebesgue norms of the maximal solution of the Magnetohydrodynamics Equations and present some criteria for global existence of solution. Thus, we can understand better on the blow-up behavior of this same solution. In addition, it is important to point out that we reach our main results by using standard techniques obtained from Navier-Stokes Equations.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488751","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 : 2018-01-01DOI: 10.4310/MAA.2018.V25.N2.A2
Tingjun Wang, Wen Zhang, Chen-Yi Zhu
. In this paper, several classical implicit finite difference schemes for solving the nonlin- ear Schr¨odinger/Gross Pitaevskii (NLS/GP) equation are revisited and analyzed. By introducing a kind of energy functionals, these schemes are proved to preserve the total energy in the discrete sense. Besides the standard energy method, a ‘cut-off’ technique and a ‘lifting’ technique are adopted to establish the optimal point-wise error estimates without any restriction on the grid ratios. Numerical results are reported to verify the theoretical analysis.
{"title":"Conservation laws and error estimates of several classical finite difference schemes for the nonlinear Schrödinger/Gross–Pitaevskii equation","authors":"Tingjun Wang, Wen Zhang, Chen-Yi Zhu","doi":"10.4310/MAA.2018.V25.N2.A2","DOIUrl":"https://doi.org/10.4310/MAA.2018.V25.N2.A2","url":null,"abstract":". In this paper, several classical implicit finite difference schemes for solving the nonlin- ear Schr¨odinger/Gross Pitaevskii (NLS/GP) equation are revisited and analyzed. By introducing a kind of energy functionals, these schemes are proved to preserve the total energy in the discrete sense. Besides the standard energy method, a ‘cut-off’ technique and a ‘lifting’ technique are adopted to establish the optimal point-wise error estimates without any restriction on the grid ratios. Numerical results are reported to verify the theoretical analysis.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489007","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 : 2018-01-01DOI: 10.4310/maa.2018.v25.n3.a4
T. Fukuda, S. Janeczko, S. Janeczko
Using J. Mather results on solutions of generic linear equations the smooth solvability of implicit differential systems is investigated. Implicit Hamiltonian systems are considered and algebraic version of J. Mather theorem was applied in this case. For the generalized Hamiltonian systems defined by P.A.M. Dirac on smooth constraints we find the corresponding Poisson-Lie algebras as a basic symplectic invariants of submanifolds in the symplectic space.
{"title":"Solvable submanifolds of tangent bundle and J. Mather generic linear equations","authors":"T. Fukuda, S. Janeczko, S. Janeczko","doi":"10.4310/maa.2018.v25.n3.a4","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n3.a4","url":null,"abstract":"Using J. Mather results on solutions of generic linear equations the smooth solvability of implicit differential systems is investigated. Implicit Hamiltonian systems are considered and algebraic version of J. Mather theorem was applied in this case. For the generalized Hamiltonian systems defined by P.A.M. Dirac on smooth constraints we find the corresponding Poisson-Lie algebras as a basic symplectic invariants of submanifolds in the symplectic space.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488940","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 : 2018-01-01DOI: 10.4310/maa.2018.v25.n4.a1
Huhe Han, T. Nishimura
. It is known that the Wulff construction is an isometry. In this paper we provide an alternative proof of this fact. Moreover, according to this result we investigate the limit of the Hausdorff distance for one-parameter families of Wulff shapes constructed by affine perturbations of dual Wulff shapes.
{"title":"Limit of the Hausdorff distance for one-parameter families of Wulff shapes constructed by affine perturbations of dual Wulff shapes","authors":"Huhe Han, T. Nishimura","doi":"10.4310/maa.2018.v25.n4.a1","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n4.a1","url":null,"abstract":". It is known that the Wulff construction is an isometry. In this paper we provide an alternative proof of this fact. Moreover, according to this result we investigate the limit of the Hausdorff distance for one-parameter families of Wulff shapes constructed by affine perturbations of dual Wulff shapes.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488951","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 : 2018-01-01DOI: 10.4310/maa.2018.v25.n4.a6
Solomon Jekel
. Presented here is a version of my talk at the Tsinghua Sanya International Mathe- matics Conference on Singularities in memory of John Mather. This article is partly expository. I will briefly recount the rise of the modern theory of foliations, describe John Mather’s contributions and then allow the discussion to lead to a report of work, old and new, on Real Analytic Gamma Structures.
{"title":"Gamma structures on surfaces","authors":"Solomon Jekel","doi":"10.4310/maa.2018.v25.n4.a6","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n4.a6","url":null,"abstract":". Presented here is a version of my talk at the Tsinghua Sanya International Mathe- matics Conference on Singularities in memory of John Mather. This article is partly expository. I will briefly recount the rise of the modern theory of foliations, describe John Mather’s contributions and then allow the discussion to lead to a report of work, old and new, on Real Analytic Gamma Structures.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489029","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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