We analyze the large deviations for a discrete energy Kac-like walk. In particular, we exhibit a path, with probability exponentially small in the number of particles, that looses energy.
{"title":"Large deviations for a binary collision model: energy evaporation","authors":"G. Basile, D. Benedetto, L. Bertini, E. Caglioti","doi":"10.3934/mine.2023001","DOIUrl":"https://doi.org/10.3934/mine.2023001","url":null,"abstract":"<abstract><p>We analyze the large deviations for a discrete energy Kac-like walk. In particular, we exhibit a path, with probability exponentially small in the number of particles, that looses energy.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42024083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider traveling waves for the Gross-Pitaevskii equation which are $ T $-periodic in each variable. We prove that if $ T $ is large enough, there exists a solution as a global minimizer of the corresponding action functional. In the subsonic case, we can use variational methods to prove the existence of a mountain-pass solution. Moreover, we show that for small $ T $ the problem admits only constant solutions.
本文研究了各变量为$ T $周期的Gross-Pitaevskii方程的行波。我们证明了如果$ T $足够大，存在一个解作为相应的作用泛函的全局最小值。在亚音速情况下，我们可以使用变分方法来证明山口解的存在性。此外，我们证明了对于小$ T $，问题只允许常数解。
{"title":"Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori","authors":"F. S'anchez, D. Ruiz","doi":"10.3934/mine.2023011","DOIUrl":"https://doi.org/10.3934/mine.2023011","url":null,"abstract":"In this paper we consider traveling waves for the Gross-Pitaevskii equation which are $ T $-periodic in each variable. We prove that if $ T $ is large enough, there exists a solution as a global minimizer of the corresponding action functional. In the subsonic case, we can use variational methods to prove the existence of a mountain-pass solution. Moreover, we show that for small $ T $ the problem admits only constant solutions.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41586738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we employ the technique developed in [2] to prove rotational symmetry for a class of Serrin-type problems for the standard Laplacian. We also discuss in some length how our strategy compares with the classical moving plane method.
{"title":"Symmetry results for Serrin-type problems in ring-shaped domains","authors":"S. Borghini","doi":"10.3934/mine.2023027","DOIUrl":"https://doi.org/10.3934/mine.2023027","url":null,"abstract":"<abstract><p>In this work, we employ the technique developed in <sup>[<xref ref-type=\"bibr\" rid=\"b2\">2</xref>]</sup> to prove rotational symmetry for a class of Serrin-type problems for the standard Laplacian. We also discuss in some length how our strategy compares with the classical moving plane method.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"65 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41305935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.
{"title":"Normal form for lower dimensional elliptic tori in Hamiltonian systems","authors":"Chiara Caracciolo","doi":"10.3934/mine.2022051","DOIUrl":"https://doi.org/10.3934/mine.2022051","url":null,"abstract":"We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45616242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $ mathbb{S}^2 $-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $ z $-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows for establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.
本文讨论了在柱面上定义的$ mathbb{S}^2 $值映射类中的dirichlet型能量泛函的全局极小值分析。该模型作为向列液晶和微磁学理论中的弯曲薄膜极限而自然出现。我们证明了最小构型是$ z $不变的，并且弱轴对称竞争类中的能量最小值实际上是轴对称的。我们的主要结果是圆柱体上的一个尖锐的庞加莱姆齐式不等式族，它允许建立一个几乎完整的能源图景。强调并讨论了对称破缺现象的存在。最后，我们提供了平面内最小化器的完整表征，它通常出现在数值模拟中，原因我们解释了。
{"title":"On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces","authors":"G. Fratta, A. Fiorenza, V. Slastikov","doi":"10.3934/mine.2023056","DOIUrl":"https://doi.org/10.3934/mine.2023056","url":null,"abstract":"The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $ mathbb{S}^2 $-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $ z $-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows for establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45698335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop and analyze a variational model for laminated paperboard. The model consists of a number of elastic sheets of a given thickness, which – at the expense of an energy per unit area – may delaminate. By providing an explicit construction for possible admissible deformations subject to boundary conditions that introduce a single bend, we discover a rich variety of energetic regimes. The regimes correspond to the experimentally observed: initial purely elastic response for small bending angle and the formation of a localized inelastic, delaminated hinge once the angle reaches a critical value. Our scaling upper bound then suggests the occurrence of several additional regimes as the angle increases. The upper bounds for the energy are partially matched by scaling lower bounds.
{"title":"Variational modeling of paperboard delamination under bending","authors":"P. Dondl, S. Conti, J. Orlik","doi":"10.3934/mine.2023039","DOIUrl":"https://doi.org/10.3934/mine.2023039","url":null,"abstract":"We develop and analyze a variational model for laminated paperboard. The model consists of a number of elastic sheets of a given thickness, which – at the expense of an energy per unit area – may delaminate. By providing an explicit construction for possible admissible deformations subject to boundary conditions that introduce a single bend, we discover a rich variety of energetic regimes. The regimes correspond to the experimentally observed: initial purely elastic response for small bending angle and the formation of a localized inelastic, delaminated hinge once the angle reaches a critical value. Our scaling upper bound then suggests the occurrence of several additional regimes as the angle increases. The upper bounds for the energy are partially matched by scaling lower bounds.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48619557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Goal of this paper is to study the following doubly nonlocal equation
begin{document} $(- Delta)^s u + mu u = (I_alpha*F(u))F'(u) quad {rm{in}};{mathbb{R}^N}qquadqquadqquadqquad ({rm{P}}) $ end{document}
in the case of general nonlinearities $ F in C^1(mathbb{R}) $ of Berestycki-Lions type, when $ N geq 2 $ and $ mu > 0 $ is fixed. Here $ (-Delta)^s $, $ s in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{alpha} $, $ alpha in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [23,61].
Goal of this paper is to study the following doubly nonlocal equation begin{document}$(- Delta)^s u + mu u = (I_alpha*F(u))F'(u) quad {rm{in}};{mathbb{R}^N}qquadqquadqquadqquad ({rm{P}})$ end{document} in the case of general nonlinearities $ F in C^1(mathbb{R}) $ of Berestycki-Lions type, when $ N geq 2 $ and $ mu > 0 $ is fixed. Here $ (-Delta)^s $, $ s in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{alpha} $, $ alpha in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [23,61].
{"title":"On fractional Schrödinger equations with Hartree type nonlinearities","authors":"S. Cingolani, Marco Gallo, Kazunaga Tanaka","doi":"10.3934/mine.2022056","DOIUrl":"https://doi.org/10.3934/mine.2022056","url":null,"abstract":"<abstract><p>Goal of this paper is to study the following doubly nonlocal equation</p>\u0000\u0000<p><disp-formula>\u0000 <label/>\u0000 <tex-math id=\"FE1\">\u0000 begin{document}\u0000$(- Delta)^s u + mu u = (I_alpha*F(u))F'(u) quad {rm{in}};{mathbb{R}^N}qquadqquadqquadqquad ({rm{P}})\u0000$\u0000 end{document}\u0000 </tex-math>\u0000</disp-formula></p>\u0000\u0000<p>in the case of general nonlinearities $ F in C^1(mathbb{R}) $ of Berestycki-Lions type, when $ N geq 2 $ and $ mu > 0 $ is fixed. Here $ (-Delta)^s $, $ s in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{alpha} $, $ alpha in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in <sup>[<xref ref-type=\"bibr\" rid=\"b23\">23</xref>,<xref ref-type=\"bibr\" rid=\"b61\">61</xref>]</sup>.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46135257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stefano Biagi, S. Dipierro, E. Valdinoci, E. Vecchi
Given a bounded open set $ Omegasubseteq{mathbb{R}}^n $, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $ Omega $. We prove that the second eigenvalue $ lambda_2(Omega) $ is always strictly larger than the first eigenvalue $ lambda_1(B) $ of a ball $ B $ with volume half of that of $ Omega $. This bound is proven to be sharp, by comparing to the limit case in which $ Omega $ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.
{"title":"A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators","authors":"Stefano Biagi, S. Dipierro, E. Valdinoci, E. Vecchi","doi":"10.3934/mine.2023014","DOIUrl":"https://doi.org/10.3934/mine.2023014","url":null,"abstract":"Given a bounded open set $ Omegasubseteq{mathbb{R}}^n $, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $ Omega $. We prove that the second eigenvalue $ lambda_2(Omega) $ is always strictly larger than the first eigenvalue $ lambda_1(B) $ of a ball $ B $ with volume half of that of $ Omega $. This bound is proven to be sharp, by comparing to the limit case in which $ Omega $ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46067236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.
{"title":"Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations","authors":"G. Gaeta, R. Kozlov, Francesco Spadaro","doi":"10.3934/mine.2022038","DOIUrl":"https://doi.org/10.3934/mine.2022038","url":null,"abstract":"We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43243818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Santo, F. Fanelli, Gabriele Sbaiz, Aneta Wr'oblewska-Kami'nska
In the present paper, we study a multiscale limit for the barotropic Navier-Stokes system with Coriolis and gravitational forces, for vanishing values of the Mach, Rossby and Froude numbers ($ {rm{Ma}} $, $ {rm{Ro}} $ and $ {rm{Fr}} $, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime $ {rm{Ma}}/{rm{Fr}}, rightarrow, 0 $, we consider scaling for the Froude number which go beyond the "critical" value $ {rm{Fr, = , sqrt{rm{Ma}}}} $. The rigorous derivation of suitable limiting systems for the various choices of the scaling is shown by means of a compensated compactness argument. Exploiting the precise structure of the gravitational force is the key to get the convergence.
{"title":"On the influence of gravity in the dynamics of geophysical flows","authors":"D. Santo, F. Fanelli, Gabriele Sbaiz, Aneta Wr'oblewska-Kami'nska","doi":"10.3934/mine.2023008","DOIUrl":"https://doi.org/10.3934/mine.2023008","url":null,"abstract":"In the present paper, we study a multiscale limit for the barotropic Navier-Stokes system with Coriolis and gravitational forces, for vanishing values of the Mach, Rossby and Froude numbers ($ {rm{Ma}} $, $ {rm{Ro}} $ and $ {rm{Fr}} $, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime $ {rm{Ma}}/{rm{Fr}}, rightarrow, 0 $, we consider scaling for the Froude number which go beyond the \"critical\" value $ {rm{Fr, = , sqrt{rm{Ma}}}} $. The rigorous derivation of suitable limiting systems for the various choices of the scaling is shown by means of a compensated compactness argument. Exploiting the precise structure of the gravitational force is the key to get the convergence.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44747753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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