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Kolmogorov variation: KAM with knobs (à la Kolmogorov) Kolmogorov变体：带旋钮的Kam
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-09-13 DOI: 10.3934/mine.2023089
M. Sansottera, Veronica Danesi
{"title":"Kolmogorov variation: KAM with knobs (à la Kolmogorov)","authors":"M. Sansottera, Veronica Danesi","doi":"10.3934/mine.2023089","DOIUrl":"https://doi.org/10.3934/mine.2023089","url":null,"abstract":"<abstract><p>In this paper we reconsider the original Kolmogorov normal form algorithm <sup>[<xref ref-type=\"bibr\" rid=\"b26\">26</xref>]</sup> with a variation on the handling of the frequencies. At difference with respect to the Kolmogorov approach, we do not keep the frequencies fixed along the normalization procedure. Besides, we select the frequencies of the final invariant torus and determine <italic>a posteriori</italic> the corresponding starting ones. In particular, we replace the classical <italic>translation step</italic> with a change of the frequencies. The algorithm is based on the original scheme of Kolmogorov, thus exploiting the fast convergence of the Newton-Kantorovich method.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44910880","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}
引用次数: 2
Interpolating estimates with applications to some quantitative symmetry results 插值估计及其在某些数量对称性结果中的应用
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-09-07 DOI: 10.3934/mine.2023002
R. Magnanini, Giorgio Poggesi
{"title":"Interpolating estimates with applications to some quantitative symmetry results","authors":"R. Magnanini, Giorgio Poggesi","doi":"10.3934/mine.2023002","DOIUrl":"https://doi.org/10.3934/mine.2023002","url":null,"abstract":"We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48073068","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}
引用次数: 10
Space-time statistics of a linear dynamical energy cascade model 线性动态能量级联模型的时空统计
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-09-01 DOI: 10.3934/mine.2023025
G. B. Apolin'ario, L. Chevillard
{"title":"Space-time statistics of a linear dynamical energy cascade model","authors":"G. B. Apolin'ario, L. Chevillard","doi":"10.3934/mine.2023025","DOIUrl":"https://doi.org/10.3934/mine.2023025","url":null,"abstract":"A linear dynamical model for the development of the turbulent energy cascade was introduced in Apolinário et al. (J. Stat. Phys., 186, 15 (2022)). This partial differential equation, randomly stirred by a forcing term which is smooth in space and delta-correlated in time, was shown to converge at infinite time towards a state of finite variance, without the aid of viscosity. Furthermore, the spatial profile of its solution gets rough, with the same regularity as a fractional Gaussian field. We here focus on the temporal behavior and derive explicit asymptotic predictions for the correlation function in time of this solution and observe that their regularity is not influenced by the spatial regularity of the problem, only by the correlation in time of the stirring contribution. We also show that the correlation in time of the solution depends on the position, contrary to its correlation in space at fixed times. We then investigate the influence of a forcing which is correlated in time on the spatial and time statistics of this equation. In this situation, while for small correlation times the homogeneous spatial statistics of the white-in-time case are recovered, for large correlation times homogeneity is broken, and a concentration around the origin of the system is observed in the velocity profiles. In other words, this fractional velocity field is a representation in one-dimension, through a linear dynamical model, of the self-similar velocity fields proposed by Kolmogorov in 1941, but only at fixed times, for a delta-correlated forcing, in which case the spatial statistics is homogeneous and rough, as expected of a turbulent velocity field. The regularity in time of turbulence, however, is not captured by this model.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45288780","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}
引用次数: 3
Poincaré inequalities and Neumann problems for the variable exponent setting 变指数集的poincar<s:1>不等式和Neumann问题
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-08-21 DOI: 10.3934/mine.2022036
D. Cruz-Uribe, Michael Penrod, S. Rodney
{"title":"Poincaré inequalities and Neumann problems for the variable exponent setting","authors":"D. Cruz-Uribe, Michael Penrod, S. Rodney","doi":"10.3934/mine.2022036","DOIUrl":"https://doi.org/10.3934/mine.2022036","url":null,"abstract":"In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45222155","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}
引用次数: 2
Spectral enclosures for the damped elastic wave equation 阻尼弹性波动方程的谱框
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-08-17 DOI: 10.3934/mine.2022052
B. Cassano, Lucrezia Cossetti, L. Fanelli
{"title":"Spectral enclosures for the damped elastic wave equation","authors":"B. Cassano, Lucrezia Cossetti, L. Fanelli","doi":"10.3934/mine.2022052","DOIUrl":"https://doi.org/10.3934/mine.2022052","url":null,"abstract":"In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lamé operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70222857","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}
引用次数: 5
Sub-exponential mixing of generalized cellular flows with bounded palenstrophy 广义元胞流的次指数混合
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-08-12 DOI: 10.3934/mine.2023006
Gianluca Crippa, Christian Schulze
{"title":"Sub-exponential mixing of generalized cellular flows with bounded palenstrophy","authors":"Gianluca Crippa, Christian Schulze","doi":"10.3934/mine.2023006","DOIUrl":"https://doi.org/10.3934/mine.2023006","url":null,"abstract":"We study the mixing properties of a passive scalar advected by an incompressible flow. We consider a class of cellular flows (more general than the class in [Crippa-Schulze M$ ^3 $AS 2017]) and show that, under the constraint that the palenstrophy is bounded uniformly in time, the mixing scale of the passive scalar cannot decay exponentially.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43557594","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}
引用次数: 1
A matrix Harnack inequality for semilinear heat equations 半线性热方程的矩阵哈纳克不等式
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-07-29 DOI: 10.3934/mine.2023003
G. Ascione, D. Castorina, G. Catino, C. Mantegazza
{"title":"A matrix Harnack inequality for semilinear heat equations","authors":"G. Ascione, D. Castorina, G. Catino, C. Mantegazza","doi":"10.3934/mine.2023003","DOIUrl":"https://doi.org/10.3934/mine.2023003","url":null,"abstract":"<abstract><p>We derive a matrix version of Li & Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type=\"bibr\" rid=\"b5\">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43724557","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}
引用次数: 1
Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow 修正Mullins-Sekerka和表面扩散流的全局存在性和稳定性
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-07-26 DOI: 10.3934/mine.2022054
Serena Della Corte, A. Diana, C. Mantegazza
{"title":"Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow","authors":"Serena Della Corte, A. Diana, C. Mantegazza","doi":"10.3934/mine.2022054","DOIUrl":"https://doi.org/10.3934/mine.2022054","url":null,"abstract":"<abstract><p>In this survey we present the state of the art about the asymptotic behavior and stability of the <italic>modified Mullins</italic>–<italic>Sekerka flow</italic> and the <italic>surface diffusion flow</italic> of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the <italic>strict stability</italic> property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently \"close\" to a smooth <italic>strictly stable critical</italic> set $ E $, both flows exist for all positive times and asymptotically \"converge\" to a translate of $ E $.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46795107","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}
引用次数: 4
A quantitative stability inequality for fractional capacities 分数容量的数量稳定性不等式
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-07-17 DOI: 10.3934/mine.2022044
E. Cinti, R. Ognibene, B. Ruffini
{"title":"A quantitative stability inequality for fractional capacities","authors":"E. Cinti, R. Ognibene, B. Ruffini","doi":"10.3934/mine.2022044","DOIUrl":"https://doi.org/10.3934/mine.2022044","url":null,"abstract":"The aim of this work is to show a non-sharp quantitative stability version of the fractional isocapacitary inequality. In particular, we provide a lower bound for the isocapacitary deficit in terms of the Fraenkel asymmetry. In addition, we provide the asymptotic behaviour of the $ s $-fractional capacity when $ s $ goes to $ 1 $ and the stability of our estimate with respect to the parameter $ s $.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41316154","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}
引用次数: 1
Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains 环空和外域hsamnon型全非线性方程的径向解
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2021-07-12 DOI: 10.3934/mine.2022055
L. Maia, Gabrielle Nornberg
{"title":"Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains","authors":"L. Maia, Gabrielle Nornberg","doi":"10.3934/mine.2022055","DOIUrl":"https://doi.org/10.3934/mine.2022055","url":null,"abstract":"In this note we study existence of positive radial solutions in annuli and exterior domains for a class of nonlinear equations driven by Pucci extremal operators subject to a Hénon type weight. Our approach is based on the shooting method applied to the corresponding ODE problem, energy arguments, and the associated flow of an autonomous quadratic dynamical system.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46742718","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}
引用次数: 2
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