We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $varepsilon to 0$, the Froude number proportional to $sqrt{varepsilon}$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $varepsilon to 0$. The limit velocity field is solenoidal and satisfies the incompressible Oberbeck-Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we apply the spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.
研究了当马赫数与小参数$varepsilon to 0$成正比，弗鲁德数与$sqrt{varepsilon}$成正比，流体占据较大区域，粗糙表面空间障碍物变化为$varepsilon to 0$时，Navier-Stokes-Fourier系统解的渐近极限。极限速度场是螺线形的，满足不可压缩的Oberbeck-Boussinesq近似。我们的研究基于弱解方法，为了达到对流项的极限，我们应用了控制声波运动的相关波传播子(诺伊曼-拉普拉斯算子)的频谱分析。
{"title":"Stability with respect to domain of the low Mach number limit of compressible heat-conducting viscous fluid","authors":"Aneta Wr'oblewska-Kami'nska","doi":"10.5817/am2023-2-231","DOIUrl":"https://doi.org/10.5817/am2023-2-231","url":null,"abstract":"We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $varepsilon to 0$, the Froude number proportional to $sqrt{varepsilon}$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $varepsilon to 0$. The limit velocity field is solenoidal and satisfies the incompressible Oberbeck-Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we apply the spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"16 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74666693","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}
M. Geyer, Jan Hausmann, Konrad Kitzing, Madlyn Senkyr, S. Siegmund
Using Maxwell's mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations $operatorname{curl} mathbf{E} = -frac{partial mathbf{B}}{partial t}$, $operatorname{curl} mathbf{H} = frac{partial mathbf{D}}{partial t} + mathbf{j}$, $operatorname{div} mathbf{D} = varrho$, $operatorname{div} mathbf{B} = 0$, which together with the constituting relations $mathbf{D} = varepsilon_0 mathbf{E}$, $mathbf{B} = mu_0 mathbf{H}$, form what we call today Maxwell's equations. Main tools are the divergence, curl and gradient integration theorems and a version of Poincare's lemma formulated in vector calculus notation. Remarks on the history of the development of electrodynamic theory, quotations and references to original and secondary literature complement the paper.
{"title":"Maxwell’s equations revisited – mental imagery and mathematical symbols","authors":"M. Geyer, Jan Hausmann, Konrad Kitzing, Madlyn Senkyr, S. Siegmund","doi":"10.5817/am2023-1-47","DOIUrl":"https://doi.org/10.5817/am2023-1-47","url":null,"abstract":"Using Maxwell's mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations $operatorname{curl} mathbf{E} = -frac{partial mathbf{B}}{partial t}$, $operatorname{curl} mathbf{H} = frac{partial mathbf{D}}{partial t} + mathbf{j}$, $operatorname{div} mathbf{D} = varrho$, $operatorname{div} mathbf{B} = 0$, which together with the constituting relations $mathbf{D} = varepsilon_0 mathbf{E}$, $mathbf{B} = mu_0 mathbf{H}$, form what we call today Maxwell's equations. Main tools are the divergence, curl and gradient integration theorems and a version of Poincare's lemma formulated in vector calculus notation. Remarks on the history of the development of electrodynamic theory, quotations and references to original and secondary literature complement the paper.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"8 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88531640","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}
Nonlinear Schr"odinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schr"odinger-Newton and Gross-Pitaevskii equations with harmonic potentials.
{"title":"Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions","authors":"F. Ficek","doi":"10.5817/AM2023-1-31","DOIUrl":"https://doi.org/10.5817/AM2023-1-31","url":null,"abstract":"Nonlinear Schr\"odinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schr\"odinger-Newton and Gross-Pitaevskii equations with harmonic potentials.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"41 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82719326","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 short note we discuss 𝑁 -supersymmetric worldlines of relativistic massless particles and review the known result that physical spin- 𝑁 / 2 fields are in the first BRST cohomology group. For 𝑁 = 1 , 2 , 4, emphasis is given to particular deformations of the BRST differential, that implement either a covariant derivative for a gauge theory or a metric connection in the target space seen by the particle. In the end, we comment about the possibility of incorporating Ramond-Ramond fluxes in the background.
{"title":"Particles in the superworldline and BRST","authors":"E. Boffo","doi":"10.5817/am2022-5-259","DOIUrl":"https://doi.org/10.5817/am2022-5-259","url":null,"abstract":"In this short note we discuss 𝑁 -supersymmetric worldlines of relativistic massless particles and review the known result that physical spin- 𝑁 / 2 fields are in the first BRST cohomology group. For 𝑁 = 1 , 2 , 4, emphasis is given to particular deformations of the BRST differential, that implement either a covariant derivative for a gauge theory or a metric connection in the target space seen by the particle. In the end, we comment about the possibility of incorporating Ramond-Ramond fluxes in the background.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"58 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78688854","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}
Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.
{"title":"Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems","authors":"Philippe Laurencçot, Bogdan–Vasile Matioc","doi":"10.5817/am2023-2-201","DOIUrl":"https://doi.org/10.5817/am2023-2-201","url":null,"abstract":"Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"99 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75855334","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}
These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Snrí, Check Republic, mostly based on [Gro20] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.
{"title":"Curvature and the equivalence problem in sub-Riemannian geometry","authors":"E. Grong","doi":"10.5817/am2022-5-295","DOIUrl":"https://doi.org/10.5817/am2022-5-295","url":null,"abstract":"These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Snrí, Check Republic, mostly based on [Gro20] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"19 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78463941","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}
. For a manifold M endowed with a Legendrean (or Lagrangean) contact structure E ⊕ F ⊂ TM , we give an elementary construction of an invariant partial connection on the quotient bundle TM/F . This permits us to develop a na¨ıve version of relative tractor calculus and to construct a second order invariant differential operator, which turns out to be the first relative BGG operator induced by the partial connection.
{"title":"Elementary relative tractor calculus for Legendrean contact structures","authors":"M. Wasilewicz","doi":"10.5817/AM2022-5-209","DOIUrl":"https://doi.org/10.5817/AM2022-5-209","url":null,"abstract":". For a manifold M endowed with a Legendrean (or Lagrangean) contact structure E ⊕ F ⊂ TM , we give an elementary construction of an invariant partial connection on the quotient bundle TM/F . This permits us to develop a na¨ıve version of relative tractor calculus and to construct a second order invariant differential operator, which turns out to be the first relative BGG operator induced by the partial connection.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"38 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73734321","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 study the existence of solutions in a Banach space of boundary value problems for Caputo-Hadamard fractional differential inclusions of order r ∈ (0 , 1].
{"title":"Boundary value problems for Caputo-Hadamard fractional differential inclusions in Banach spaces","authors":"Amouria Hammou, S. Hamani, J. Henderson","doi":"10.5817/am2022-4-227","DOIUrl":"https://doi.org/10.5817/am2022-4-227","url":null,"abstract":". In this article, we study the existence of solutions in a Banach space of boundary value problems for Caputo-Hadamard fractional differential inclusions of order r ∈ (0 , 1].","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"113 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79861487","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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