We define harmonic maps between sub-Riemannian manifolds by generalizing known definitions for Riemannian manifolds. We establish conditions for when a horizontal map into a Lie group with a left-invariant metric structure is a harmonic map. We show that sub-Riemannian harmonic maps can be abnormal or normal, just as sub-Riemannian geodesics. We illustrate our study by presenting the equations for harmonic maps into the Heisenberg group.
{"title":"Harmonic maps into sub-Riemannian Lie groups","authors":"E. Grong, I. Markina","doi":"10.3934/cam.2023025","DOIUrl":"https://doi.org/10.3934/cam.2023025","url":null,"abstract":"We define harmonic maps between sub-Riemannian manifolds by generalizing known definitions for Riemannian manifolds. We establish conditions for when a horizontal map into a Lie group with a left-invariant metric structure is a harmonic map. We show that sub-Riemannian harmonic maps can be abnormal or normal, just as sub-Riemannian geodesics. We illustrate our study by presenting the equations for harmonic maps into the Heisenberg group.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132705226","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}
In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by van der Schaft and Maschke in 2002. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.
在本文中，我们证明了Hilbert空间上的一大类线性演化偏微分方程定义了一个Stokes-Dirac结构。为此，采用了边界控制系统理论。这个定义包含了在2002年van der Schaft和Maschke的开创性论文中给出的几何设置不能处理的力学问题。本文详细介绍了连续介质力学和物理学的许多算例，并特别关注几何域边界上的对偶功能空间。对于每个例子，微分算子和相关的希尔伯特复形之间的联系都被说明。
{"title":"Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint","authors":"A. Brugnoli, Ghislain Haine, D. Matignon","doi":"10.3934/cam.2023018","DOIUrl":"https://doi.org/10.3934/cam.2023018","url":null,"abstract":"In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by van der Schaft and Maschke in 2002. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124313947","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}
We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $ K $-contact structures and allow a new look at the classical theory. We characterize weak $ K $-contact manifolds among all weak contact metric manifolds using the property well known for $ K $-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $ K $-contact manifold. We also find sufficient conditions for a weak $ K $-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.
{"title":"Generalized Ricci solitons and Einstein metrics on weak $ K $-contact manifolds","authors":"V. Rovenski","doi":"10.3934/cam.2023010","DOIUrl":"https://doi.org/10.3934/cam.2023010","url":null,"abstract":"We study so-called \"weak\" metric structures on a smooth manifold, which generalize the metric contact and $ K $-contact structures and allow a new look at the classical theory. We characterize weak $ K $-contact manifolds among all weak contact metric manifolds using the property well known for $ K $-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $ K $-contact manifold. We also find sufficient conditions for a weak $ K $-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133710973","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}
We detail a calculation of the second order normal form of the Stark effect Hamiltonian after regularization, using the Kustaanheimo-Stiefel mapping. After reduction, we obtain an integrable two degree of freedom system on $ S^2_h times S^2_h $, which we reduce again to obtain a one degree of freedom Hamiltonian system.
{"title":"Normalization and reduction of the Stark Hamiltonian","authors":"R. Cushman","doi":"10.3934/cam.2023022","DOIUrl":"https://doi.org/10.3934/cam.2023022","url":null,"abstract":"We detail a calculation of the second order normal form of the Stark effect Hamiltonian after regularization, using the Kustaanheimo-Stiefel mapping. After reduction, we obtain an integrable two degree of freedom system on $ S^2_h times S^2_h $, which we reduce again to obtain a one degree of freedom Hamiltonian system.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127935999","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}
In this paper, we establish two conclusions about the continuous dependence on the initial data of the global solution to the initial boundary value problem of a porous elastic system for the linear damping case and the nonlinear damping case, respectively, which reflect the decay property of the dissipative system. Additionally, we estimate the lower bound of the blowup time at the arbitrary positive initial energy for nonlinear damping case.
{"title":"Continuous dependence on initial data and high energy blowup time estimate for porous elastic system","authors":"Jiangbo Han, Runzhang Xu, Chao Yang","doi":"10.3934/cam.2023012","DOIUrl":"https://doi.org/10.3934/cam.2023012","url":null,"abstract":"In this paper, we establish two conclusions about the continuous dependence on the initial data of the global solution to the initial boundary value problem of a porous elastic system for the linear damping case and the nonlinear damping case, respectively, which reflect the decay property of the dissipative system. Additionally, we estimate the lower bound of the blowup time at the arbitrary positive initial energy for nonlinear damping case.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125627295","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}
We present an approach to Lyapunov theorems about a center for germs of analytic vector fields based on the Poincaré–Dulac and Birkhoff normal forms. Besides new proofs of three Lyapunov theorems, we prove their generalization: if the Poincaré–Dulac normal form indicates the existence of a family of periodic solutions, then such a family really exists. We also present new proofs of Weinstein and Moser theorems about lower bounds for the number of families of periodic solutions; here, besides the normal forms, some topological tools are used, i.e., the Poincaré–Hopf formula and the Lusternik–Schnirelmann category on weighted projective spaces.
{"title":"Normal forms, invariant manifolds and Lyapunov theorems","authors":"H. Zoladek","doi":"10.3934/cam.2023016","DOIUrl":"https://doi.org/10.3934/cam.2023016","url":null,"abstract":"We present an approach to Lyapunov theorems about a center for germs of analytic vector fields based on the Poincaré–Dulac and Birkhoff normal forms. Besides new proofs of three Lyapunov theorems, we prove their generalization: if the Poincaré–Dulac normal form indicates the existence of a family of periodic solutions, then such a family really exists. We also present new proofs of Weinstein and Moser theorems about lower bounds for the number of families of periodic solutions; here, besides the normal forms, some topological tools are used, i.e., the Poincaré–Hopf formula and the Lusternik–Schnirelmann category on weighted projective spaces.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116842112","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}
In this paper, we consider a Schrödinger operator $ L = -Delta_{mathbb{H}}+V $ on the stratified Lie group $ mathbb{H} $. First, we establish fractional heat kernel estimates related to $ L^{beta} $, $ betain(0, 1) $. By utilizing kernel estimations and the fractional Carleson measure, we are able to derive a characterization of the Campanato type space $ BMO_{L}^{v}(mathbb{H}) $. Second, we demonstrate that both Littlewood-Paley $ {bf g} $-functions and area functions are bounded on $ BMO^{v}_{L}(mathbb{H}) $. Finally, we also obtain that the above square functions are bounded on the Morrey space $ L^{gamma, theta}_{p, kappa}(mathbb{H}) $ and the weak Morrey space $ WL^{gamma, theta}_{1, kappa}(mathbb{H}) $, respectively.
{"title":"Boundedness of square functions related with fractional Schrödinger semigroups on stratified Lie groups","authors":"Zhiyong Wang, Kai Zhao, Pengtao Li, Yu Liu","doi":"10.3934/cam.2023020","DOIUrl":"https://doi.org/10.3934/cam.2023020","url":null,"abstract":"In this paper, we consider a Schrödinger operator $ L = -Delta_{mathbb{H}}+V $ on the stratified Lie group $ mathbb{H} $. First, we establish fractional heat kernel estimates related to $ L^{beta} $, $ betain(0, 1) $. By utilizing kernel estimations and the fractional Carleson measure, we are able to derive a characterization of the Campanato type space $ BMO_{L}^{v}(mathbb{H}) $. Second, we demonstrate that both Littlewood-Paley $ {bf g} $-functions and area functions are bounded on $ BMO^{v}_{L}(mathbb{H}) $. Finally, we also obtain that the above square functions are bounded on the Morrey space $ L^{gamma, theta}_{p, kappa}(mathbb{H}) $ and the weak Morrey space $ WL^{gamma, theta}_{1, kappa}(mathbb{H}) $, respectively.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130628788","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}
In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.
{"title":"Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials","authors":"Huiyang Xu","doi":"10.3934/cam.2023008","DOIUrl":"https://doi.org/10.3934/cam.2023008","url":null,"abstract":"In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121502417","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}
We assign a Riemannian metric to a system of nonlinear equations that describe the one-dimensional propagation of long magnetoacoustic waves (also called magnetosonic waves) in a cold plasma under the inference of a transverse magnetic field. The metric, which in general is expressed in terms of the density of the plasma and its speed across the magnetic field, when specialized to a particular solution of the nonlinear system (the Gurevich-Krylov (G-K) solution) is mapped explicitly to a Jackiw-Teitelboim (J-T) black hole metric, which is the main result. Dilaton fields, constructed from data involved in the G-K solution, are presented - which with the plasma metric provide for elliptic function solutions of the J-T equations of motion in 2d dilaton gravity. A correspondence between solutions of the nonlinear plasma system (whose Galilean invariance is also established) and certain solutions of a resonant nonlinear Schrödinger equation is set up, along with some other general background material to render an expository tone in the presentation.
{"title":"From a magnetoacoustic system to a J-T black hole: A little trip down memory lane","authors":"F. Williams","doi":"10.3934/cam.2023017","DOIUrl":"https://doi.org/10.3934/cam.2023017","url":null,"abstract":"We assign a Riemannian metric to a system of nonlinear equations that describe the one-dimensional propagation of long magnetoacoustic waves (also called magnetosonic waves) in a cold plasma under the inference of a transverse magnetic field. The metric, which in general is expressed in terms of the density of the plasma and its speed across the magnetic field, when specialized to a particular solution of the nonlinear system (the Gurevich-Krylov (G-K) solution) is mapped explicitly to a Jackiw-Teitelboim (J-T) black hole metric, which is the main result. Dilaton fields, constructed from data involved in the G-K solution, are presented - which with the plasma metric provide for elliptic function solutions of the J-T equations of motion in 2d dilaton gravity. A correspondence between solutions of the nonlinear plasma system (whose Galilean invariance is also established) and certain solutions of a resonant nonlinear Schrödinger equation is set up, along with some other general background material to render an expository tone in the presentation.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131455472","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}
The clamped plate problem describes the vibration of a clamped plate in the classical elastic mechanics, and the Xin-Laplacian is an important elliptic operator for understanding the geometric structure of translators of mean curvature flow(MCF for short). In this article, we investigate the clamped plate problem of the bi-Xin-Laplacian on Riemannian manifolds isometrically immersed in the Euclidean space. On one hand, we obtain some eigenvalue inequalities of the bi-Xin-Laplacian on some important Riemannian manifolds admitting some special functions. Let us emphasize that, this class of manifolds contains some interesting examples: Cartan-Hadamard manifolds, some types of warp product manifolds and homogenous spaces. On the other hand, we also consider the eigenvalue problem of the bi-Xin-Laplacian on the cylinders and obtain an eigenvalue inequality. In particular, we can give an estimate for the lower order eigenvalues on the cylinders.
{"title":"Eigenvalues of the bi-Xin-Laplacian on complete Riemannian manifolds","authors":"Xiaotian Hao, Lingzhong Zeng","doi":"10.3934/cam.2023009","DOIUrl":"https://doi.org/10.3934/cam.2023009","url":null,"abstract":"The clamped plate problem describes the vibration of a clamped plate in the classical elastic mechanics, and the Xin-Laplacian is an important elliptic operator for understanding the geometric structure of translators of mean curvature flow(MCF for short). In this article, we investigate the clamped plate problem of the bi-Xin-Laplacian on Riemannian manifolds isometrically immersed in the Euclidean space. On one hand, we obtain some eigenvalue inequalities of the bi-Xin-Laplacian on some important Riemannian manifolds admitting some special functions. Let us emphasize that, this class of manifolds contains some interesting examples: Cartan-Hadamard manifolds, some types of warp product manifolds and homogenous spaces. On the other hand, we also consider the eigenvalue problem of the bi-Xin-Laplacian on the cylinders and obtain an eigenvalue inequality. In particular, we can give an estimate for the lower order eigenvalues on the cylinders.","PeriodicalId":233941,"journal":{"name":"Communications in Analysis and Mechanics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124517312","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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