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On tidal energy in Newtonian two-body motion with infinite initial separation 关于牛顿二体运动中无限初始分离的潮汐能
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a14
Shuang Miao, Lan Zhang
{"title":"On tidal energy in Newtonian two-body motion with infinite initial separation","authors":"Shuang Miao, Lan Zhang","doi":"10.4310/pamq.2024.v20.n4.a14","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a14","url":null,"abstract":"In $href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ we have studied the dynamics of tidal energy in Newtonian two-body motion and how it affects the center-of-mass orbit of two identical gravitating fluid bodies. It is shown in $href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ that for a class of initial configuration, the tidal energy caused by the deformation of boundaries of two fluid bodies can be made arbitrarily large relative to the positive conserved total energy of the entire system. This reveals the possibility that the center-of-mass orbit, which is unbounded initially, may become bounded during the evolution. This result in $href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ is based on a quantitative relation between the tidal energy and the distance of two bodies. However, this relation only holds when the two-body distance are within multiples of the first closest approach, due to the fact that initially the tidal energy vanishes but the two-body distance is finite. In this work, based on the a priori estimates established in $href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$, we construct a solution to the same two-body problem as in $href{https://doi.org/10.48550/arXiv.1708.04307}{[8]}$ but with infinite initial separation. Therefore the above mentioned quantitative relation holds during the entire evolution up to the first closest approach.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743349","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}
引用次数: 0
Semi-global controllability of a geometric wave equation 几何波方程的半全局可控性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a9
Joachim Krieger, Shengquan Xiang
{"title":"Semi-global controllability of a geometric wave equation","authors":"Joachim Krieger, Shengquan Xiang","doi":"10.4310/pamq.2024.v20.n4.a9","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a9","url":null,"abstract":"We prove the semi-global controllability and stabilization of the $(1 + 1)$-dimensional wave maps equation with spatial domain $mathbb{S}^1$ and target $mathbb{S}^k$. First, we show that damping stabilizes the system when the energy is strictly below the threshold $2pi$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k = 1$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743343","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}
引用次数: 0
Supertranslation invariance of angular momentum at null infinity in double null gauge 双空规中无穷远处角动量的超平移不变性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a5
Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau
{"title":"Supertranslation invariance of angular momentum at null infinity in double null gauge","authors":"Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau","doi":"10.4310/pamq.2024.v20.n4.a5","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a5","url":null,"abstract":"The supertranslation invariance of the Chen-Wang-Yau (CWY) angular momentum in the Bondi-Sachs formalism/gauge was ascertained by the authors in [$href{https://doi.org/10.1007/s00220-021-04053-7}{12}$, $href{https://dx.doi.org/10.4310/ATMP.2021.v25.n3.a4}{13}$]. In this article, we study the corresponding problem in the double null gauge. In particular, supertranslation ambiguity of this gauge is identified and the CWY angular momentum is proven to be free of this ambiguity. A similar result is obtained for the CWY center of mass integral.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746154","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}
引用次数: 0
Null Penrose inequality in a perturbed Schwarzschild spacetime 扰动施瓦兹柴尔德时空中的彭罗斯零不等式
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a6
Pengyu Le
{"title":"Null Penrose inequality in a perturbed Schwarzschild spacetime","authors":"Pengyu Le","doi":"10.4310/pamq.2024.v20.n4.a6","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a6","url":null,"abstract":"In this paper, we review the proof of the null Penrose inequality in a perturbed Schwarzschild spacetime. The null Penrose inequality conjectures that, on an incoming null hypersurface, the Hawking mass of the outmost marginally trapped surface is not greater than the Bondi mass at past null infinity. An approach to prove the null Penrose inequality is to construct a foliation on the null hypersurface starting from the marginally trapped surface to past null infinity, on which the Hawking mass is monotonically nondecreasing. However to achieve a proof, there arises an obstacle on the asymptotic geometry of the foliation at past null infinity. In order to overcome this obstacle, Christodoulou and Sauter proposed a strategy by varying the hypersurface to search for another null hypersurface where asymptotic geometry of the foliation becomes round. This strategy leads us to study the perturbation of null hypersurfaces systematically. Applying the perturbation theory of null hypersurfaces in a perturbed Schwarzschild spacetime, we carry out the strategy of Christodoulou and Sauter successfully. We find a one-parameter family of null hypersurfaces on which the null Penrose inequality holds. This paper gives a overview of our proof.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743340","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}
引用次数: 0
On the non-existence of trapped surfaces under low-regularity bounds 论低规则性约束下陷阱曲面的不存在性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a1
Jonathan Luk, Georgios Moschidis
{"title":"On the non-existence of trapped surfaces under low-regularity bounds","authors":"Jonathan Luk, Georgios Moschidis","doi":"10.4310/pamq.2024.v20.n4.a1","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a1","url":null,"abstract":"The emergence of trapped surfaces in solutions to the Einstein field equations is intimately tied to the well-posedness properties of the corresponding Cauchy problem in the low regularity regime. In this paper, we study the question of existence of trapped surfaces already at the level of the initial hypersurface when the scale invariant size of the Cauchy data is assumed to be bounded. Our main theorem states that no trapped surfaces can exist initially when the Cauchy data are close to the data induced on a spacelike hypersurface of Minkowski spacetime (not necessarily a flat hyperplane) in the Besov $B^{3/2}{2,1}$ norm. We also discuss the question of extending the above result to the case when merely smallness in $H^{3/2}$ is assumed.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743351","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}
引用次数: 0
Gravitational solitons and complete Ricci flat Riemannian manifolds of infinite topological type 引力孤子与无限拓扑类型的完整里奇平坦黎曼流形
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a12
Marcus Khuri, Martin Reiris, Gilbert Weinstein, Sumio Yamada
{"title":"Gravitational solitons and complete Ricci flat Riemannian manifolds of infinite topological type","authors":"Marcus Khuri, Martin Reiris, Gilbert Weinstein, Sumio Yamada","doi":"10.4310/pamq.2024.v20.n4.a12","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a12","url":null,"abstract":"We present several new space-periodic solutions of the static vacuum Einstein equations in higher dimensions, both with and without black holes, having Kasner asymptotics. These latter solutions are referred to as gravitational solitons. Further partially compactified solutions are also obtained by taking appropriate quotients, and the topologies are computed explicitly in terms of connected sums of products of spheres. In addition, it is shown that there is a correspondence, via Wick rotation, between the spacelike slices of the solitons and black hole solutions in one dimension less. As a corollary, the solitons give rise to complete Ricci flat Riemannian manifolds of infinite topological type and generic holonomy, in dimensions $4$ and higher.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743347","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}
引用次数: 0
Masses at null infinity for Einstein's equations in harmonic coordinates 谐波坐标下爱因斯坦方程空无穷远处的质量
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a3
Lili He, Hans Lindblad
{"title":"Masses at null infinity for Einstein's equations in harmonic coordinates","authors":"Lili He, Hans Lindblad","doi":"10.4310/pamq.2024.v20.n4.a3","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a3","url":null,"abstract":"In this work, we give a complete picture of how to, in a direct simple way, define the mass at null infinity in harmonic coordinates in three different ways that we show satisfy the Bondi mass loss law. The first and second way involve only the limit of metric (Trautman mass) respectively the null second fundamental forms along asymptotically characteristic surfaces (asymptotic Hawking mass) that only depend on the ADM mass. The last involves construction of special characteristic coordinates at null infinity (Bondi mass). The results here rely on asymptotics of the metric derived in $href{https://doi.org/10.1007/s00220-017-2876-z}{[27]}$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743339","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}
引用次数: 0
Remark on the nonlinear stability of Minkowski spacetime: a rigidity theorem 关于闵科夫斯基时空非线性稳定性的备注：刚性定理
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a7
Jin Jia, Pin Yu
{"title":"Remark on the nonlinear stability of Minkowski spacetime: a rigidity theorem","authors":"Jin Jia, Pin Yu","doi":"10.4310/pamq.2024.v20.n4.a7","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a7","url":null,"abstract":"In the framework of the nonlinear stability of Minkowski spacetime, we show that if the radiation field of the curvature tensor vanishes, the spacetime must be flat.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743342","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}
引用次数: 0
Derivation of the half-wave maps equation from Calogero–Moser spin systems 从卡洛吉罗-莫泽自旋系统推导半波图方程
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a10
Enno Lenzmann, Jérémy Sok
{"title":"Derivation of the half-wave maps equation from Calogero–Moser spin systems","authors":"Enno Lenzmann, Jérémy Sok","doi":"10.4310/pamq.2024.v20.n4.a10","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a10","url":null,"abstract":"We prove that the energy-critical half-wave maps equation[$partial_t mathbf {S} = mathbf {S} times |nabla |mathbf {S}, quad (mathit{t}, mathit{x}) in mathbb R times mathbb T$]arises as an effective equation in the continuum limit of completely integrable Calogero–Moser classical spin systems with inverse square $1/r^2$ interactions on the circle. We study both the convergence to global-in-time weak solutions in the energy class as well as short-time strong solutions of higher regularity. The proofs are based on Fourier methods and suitable discrete analogues of fractional Leibniz rules and Kato–Ponce–Vega commutator estimates.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743344","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}
引用次数: 0
Covariant guiding laws for fields 场的协变指导定律
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a13
Maaneli Derakhshani, Michael K.-H. Kiessling, A. Shadi Tahvildar-Zadeh
{"title":"Covariant guiding laws for fields","authors":"Maaneli Derakhshani, Michael K.-H. Kiessling, A. Shadi Tahvildar-Zadeh","doi":"10.4310/pamq.2024.v20.n4.a13","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a13","url":null,"abstract":"After reviewing the passage from classical Hamilton–Jacobi formulation of non-relativistic point-particle dynamics to the non-relativistic quantum dynamics of point particles whose motion is guided by a wave function that satisfies Schrödinger’s or Pauli’s equation, we study the analogous question for the Lorentz-covariant dynamics of fields on spacelike slices of spacetime. We establish a relationship, between the DeDonder–Weyl–Christodoulou formulation of covariant Hamilton–Jacobi equations for the classical field evolution, and the Lorentz-covariant Dirac-type wave equation proposed by Kanatchikov amended by our proposed guiding equation for such fields. We show that Kanatchikov’s equation is well-posed and generally solvable, and we establish the correspondence between plane-wave solutions of Kanatchikov’s equation and solutions of the covariant Hamilton–Jacobi equations ofDeDonder–Weyl–Christodoulou. We propose a covariant guiding law for the temporal evolution of fields defined on constant time slices of spacetime, and show that it yields, at each spacetime point, the existence of a finite measure on the space of field values at that point that is equivariant with respect to the flow induced by the solution of Kanatchikov’s equation that is guiding the actual field, so long as it is a plane-wave solution. We show that our guiding law is local in the sense of Einstein’s special relativity, and therefore it cannot be used to analyze Bell-type experiments. We conclude by suggesting directions to be explored in future research.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743348","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}
引用次数: 0
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