Hyunjin Ahn, Seung‐Yeal Ha, Doheon Kim, F. Schlöder, Woojoo Shim
We study the mean-field limit of the Cucker-Smale (C-S) model for flocking on complete smooth Riemannian manifolds. For this, we first formally derive the kinetic manifold C-S model on manifolds using the BBGKY hierarchy and derive several a priori estimates on emergent dynamics. Then, we present a rigorous mean-field limit from the particle model to the corresponding kinetic model by using the generalized particle-in-cell method. As a byproduct of our rigorous mean-field limit estimate, we also establish a global existence of a measure-valued solution for the derived kinetic model. Compared to the corresponding results on R d mathbb {R}^d , our procedure requires additional assumption on holonomy and proper a priori bound on the derivative of parallel transports. As a concrete example, we verify that hyperbolic space H d mathbb {H}^d satisfies our proposed standing assumptions.
研究了完全光滑黎曼流形上簇的cucker - small (C-S)模型的平均场极限。为此，我们首先使用BBGKY层次正式推导了流形上的动力学流形C-S模型，并推导了对紧急动力学的几个先验估计。然后，利用广义胞内粒子法给出了从粒子模型到相应动力学模型的严格平均场极限。作为我们严格的平均场极限估计的副产品，我们还建立了导出的动力学模型的测量值解的全局存在性。与R d mathbb {R}^d上的相应结果相比，我们的方法需要额外的完整假设和平行输运导数的固有先验界。作为一个具体的例子，我们验证了双曲空间H d mathbb {H}^d满足我们提出的常值假设。
{"title":"The mean-field limit of the Cucker-Smale model on complete Riemannian manifolds","authors":"Hyunjin Ahn, Seung‐Yeal Ha, Doheon Kim, F. Schlöder, Woojoo Shim","doi":"10.1090/qam/1613","DOIUrl":"https://doi.org/10.1090/qam/1613","url":null,"abstract":"We study the mean-field limit of the Cucker-Smale (C-S) model for flocking on complete smooth Riemannian manifolds. For this, we first formally derive the kinetic manifold C-S model on manifolds using the BBGKY hierarchy and derive several a priori estimates on emergent dynamics. Then, we present a rigorous mean-field limit from the particle model to the corresponding kinetic model by using the generalized particle-in-cell method. As a byproduct of our rigorous mean-field limit estimate, we also establish a global existence of a measure-valued solution for the derived kinetic model. Compared to the corresponding results on \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 d\u0000 \u0000 mathbb {R}^d\u0000 \u0000\u0000, our procedure requires additional assumption on holonomy and proper a priori bound on the derivative of parallel transports. As a concrete example, we verify that hyperbolic space \u0000\u0000 \u0000 \u0000 \u0000 H\u0000 \u0000 d\u0000 \u0000 mathbb {H}^d\u0000 \u0000\u0000 satisfies our proposed standing assumptions.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45905864","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}
This paper identifies a non-(or /iso-)thermal variant of Ruggeri’s 1983 formulation of viscous heat-conductive fluid dynamics as a hyperbolic system of balance laws and shows that both the original model and this variant have (a) time-asymptotically stable equilibria and (b) principal parts deriving from a protopotential: a single scalar function that induces the temporospatial flux as an appropriate part of its Hessian.
{"title":"Time-asymptotic stability for first-order symmetric hyperbolic systems of balance laws in dissipative compressible fluid dynamics","authors":"H. Freistühler","doi":"10.1090/qam/1620","DOIUrl":"https://doi.org/10.1090/qam/1620","url":null,"abstract":"This paper identifies a non-(or /iso-)thermal variant of Ruggeri’s 1983 formulation of viscous heat-conductive fluid dynamics as a hyperbolic system of balance laws and shows that both the original model and this variant have (a) time-asymptotically stable equilibria and (b) principal parts deriving from a protopotential: a single scalar function that induces the temporospatial flux as an appropriate part of its Hessian.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45557377","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}
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Large-time behavior of compressible polytropic fluids and nonlinear Schrödinger equation Rémi Carles, Kleber Carrapatoso, Matthieu Hillairet
{"title":"Large-time behavior of compressible polytropic fluids and nonlinear Schrödinger equation","authors":"R. Carles, K. Carrapatoso, M. Hillairet","doi":"10.1090/qam/1618","DOIUrl":"https://doi.org/10.1090/qam/1618","url":null,"abstract":"HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Large-time behavior of compressible polytropic fluids and nonlinear Schrödinger equation Rémi Carles, Kleber Carrapatoso, Matthieu Hillairet","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44756620","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}
This is our second work in the series about constructing boundary conditions for hyperbolic relaxation approximations. The present work is concerned with the one-dimensional linearized Jin-Xin relaxation model, a convenient approximation of hyperbolic conservation laws, with non-characteristic boundaries. Assume that proper boundary conditions are given for the conservation laws. We construct boundary conditions for the relaxation model with the expectation that the resultant initial-boundary-value problems are approximations to the given conservation laws with the boundary conditions. The constructed boundary conditions are highly non-unique. Their satisfaction of the generalized Kreiss condition is analyzed. The compatibility with initial data is studied. Furthermore, by resorting to a formal asymptotic expansion, we prove the effectiveness of the approximations.
{"title":"Construction of boundary conditions for hyperbolic relaxation approximations II: Jin-Xin relaxation model","authors":"Xiaxia Cao, W. Yong","doi":"10.1090/qam/1627","DOIUrl":"https://doi.org/10.1090/qam/1627","url":null,"abstract":"This is our second work in the series about constructing boundary conditions for hyperbolic relaxation approximations. The present work is concerned with the one-dimensional linearized Jin-Xin relaxation model, a convenient approximation of hyperbolic conservation laws, with non-characteristic boundaries. Assume that proper boundary conditions are given for the conservation laws. We construct boundary conditions for the relaxation model with the expectation that the resultant initial-boundary-value problems are approximations to the given conservation laws with the boundary conditions. The constructed boundary conditions are highly non-unique. Their satisfaction of the generalized Kreiss condition is analyzed. The compatibility with initial data is studied. Furthermore, by resorting to a formal asymptotic expansion, we prove the effectiveness of the approximations.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45801407","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}
<p>We estimate the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) local distance between piecewise constant solutions to the Cauchy problem of conservation laws and propose a shock admissibility condition for having an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> local contraction of such solutions. Moreover, as an application, we prove that there exist <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> locally contractive solutions on some set of initial functions, to the Cauchy problem of conservation laws with convex or concave flux functions. As a result, for conservation laws with convex or concave flux functions, we see that rarefaction waves have an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">qgeq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) local contraction and shock waves have an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript r"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation en
{"title":"An 𝐿^{𝑝} shock admissibility condition for conservation laws","authors":"Hiroki Ohwa","doi":"10.1090/qam/1610","DOIUrl":"https://doi.org/10.1090/qam/1610","url":null,"abstract":"<p>We estimate the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p>0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>) local distance between piecewise constant solutions to the Cauchy problem of conservation laws and propose a shock admissibility condition for having an <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> local contraction of such solutions. Moreover, as an application, we prove that there exist <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> locally contractive solutions on some set of initial functions, to the Cauchy problem of conservation laws with convex or concave flux functions. As a result, for conservation laws with convex or concave flux functions, we see that rarefaction waves have an <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript q\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>q</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^q</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q greater-than-or-equal-to 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">qgeq 1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>) local contraction and shock waves have an <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript r\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>r</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation en","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47469321","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}
In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. This cavitation model consists of a coupled problem between the compressible Reynolds PDE (that describes the flow) and the Rayleigh-Plesset ODE (that describes micro-bubbles evolution). Very few theoretical results exist in the mathematical literature about such couple problems. A complete form including bubbles convection is studied here. Local times existence results are proved based on the semi group theory. Stability theorems are obtained in a particular case.
{"title":"About a cavitation model including bubbles in thin film lubrication taking convection into account","authors":"G. Bayada, I. Ciuperca","doi":"10.1090/qam/1609","DOIUrl":"https://doi.org/10.1090/qam/1609","url":null,"abstract":"In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. This cavitation model consists of a coupled problem between the compressible Reynolds PDE (that describes the flow) and the Rayleigh-Plesset ODE (that describes micro-bubbles evolution). Very few theoretical results exist in the mathematical literature about such couple problems. A complete form including bubbles convection is studied here. Local times existence results are proved based on the semi group theory. Stability theorems are obtained in a particular case.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43443365","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}
In this paper, we consider the linear stability of traveling wave solutions for one-dimensional compressible isentropic Navier-Stokes equations with Maxwell’s Law. The global stability of traveling wave solution is established with shock-profile initial data for the linearized system. Anti-derivative and some delicate energy methods are explored to get the desired results. Moreover, the relaxation limit of traveling wave solution is also obtained.
{"title":"Linear stability of viscous shock wave for 1-D compressible Navier-Stokes equations with Maxwell’s law","authors":"Yuxi Hu, Zhao Wang","doi":"10.1090/qam/1608","DOIUrl":"https://doi.org/10.1090/qam/1608","url":null,"abstract":"In this paper, we consider the linear stability of traveling wave solutions for one-dimensional compressible isentropic Navier-Stokes equations with Maxwell’s Law. The global stability of traveling wave solution is established with shock-profile initial data for the linearized system. Anti-derivative and some delicate energy methods are explored to get the desired results. Moreover, the relaxation limit of traveling wave solution is also obtained.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44084759","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}
The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation H(x,∇u)=0,H(x,nabla u)=0, coupled with non-zero Dirichlet condition u=gu=g on ∂Ωpartial Omega. In this article, the Hamiltonian HH is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.
{"title":"Beckmann-type problem for degenerate Hamilton-Jacobi equations","authors":"Hamza Ennaji, N. Igbida, Van Thanh Nguyen","doi":"10.1090/qam/1606","DOIUrl":"https://doi.org/10.1090/qam/1606","url":null,"abstract":"<p>The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H left-parenthesis x comma nabla u right-parenthesis equals 0 comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H(x,nabla u)=0,</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> coupled with non-zero Dirichlet condition <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u equals g\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>g</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">u=g</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential normal upper Omega\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">partial Omega</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this article, the Hamiltonian <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.</p>","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47607388","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}
The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.
{"title":"Symmetry group of the equiangular cubed sphere","authors":"Jean-Baptiste Bellet","doi":"10.1090/qam/1604","DOIUrl":"https://doi.org/10.1090/qam/1604","url":null,"abstract":"The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45222240","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}
A classical problem from potential theory (a point source inside a long rigid tube) is revisited. It has an extensive literature but its resolution is not straightforward: standard approaches lead to divergent integrals or require the discarding of infinite constants. We show that the problem can be solved rigorously using classical methods.
{"title":"On Green’s function for Laplace’s equation in a rigid tube","authors":"P. Martin","doi":"10.1090/qam/1603","DOIUrl":"https://doi.org/10.1090/qam/1603","url":null,"abstract":"A classical problem from potential theory (a point source inside a long rigid tube) is revisited. It has an extensive literature but its resolution is not straightforward: standard approaches lead to divergent integrals or require the discarding of infinite constants. We show that the problem can be solved rigorously using classical methods.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46204646","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}
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