In this paper we reconsider the original Kolmogorov normal form algorithm [26] 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 a posteriori the corresponding starting ones. In particular, we replace the classical translation step 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.
{"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}
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.
{"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}
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.
{"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}
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)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.
在较早的一篇论文中，Cruz-Uribe, Rodney和Rosta证明了加权poincar不等式与一类退化的$ p $-拉普拉斯算子相关的Neumann问题弱解的存在性之间的等价性。本文证明了变指数空间中的poincar不等式与退化的$ {p(cdot)} $-拉普拉斯方程解之间的类似等价性，这是一个具有非标准生长条件的非线性椭圆方程。
{"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}
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.
{"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}
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 $AS 2017]) and show that, under the constraint that the palenstrophy is bounded uniformly in time, the mixing scale of the passive scalar cannot decay exponentially.
{"title":"Sub-exponential mixing of generalized cellular flows with bounded palenstrophy","authors":"Gianluca Crippa, Christian Schulze","doi":"10.3934/mine.2023006","DOIUrl":"https://doi.org/10.3934/mine.2023006","url":null,"abstract":"We study the mixing properties of a passive scalar advected by an incompressible flow. We consider a class of cellular flows (more general than the class in [Crippa-Schulze M$ ^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}
G. Ascione, D. Castorina, G. Catino, C. Mantegazza
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 [5] 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.
{"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}
In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins–Sekerka flow and the surface diffusion flow 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 strict stability 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 strictly stable critical set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.
本文主要介绍了E. Acerbi、N. Fusco、V. Julin和M. Morini等人关于修正Mullins-Sekerka流和光滑集表面扩散流的渐近行为和稳定性的最新研究进展。首先详细讨论了体积约束下的非局部区域泛函的性质，其中两种流是适当范数下的梯度流，特别地，我们定义了这种泛函的临界集的严格稳定性性质，并证明了它是在任意维W^{2, p} $ -摄动下极小性的充分必要条件。然后，我们证明，在维2和维3中，对于足够“接近”光滑严格稳定临界集$ E $的初始集，这两个流在所有正时间都存在，并且渐近地“收敛”到$ E $的平移。
{"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}
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 $.
这项工作的目的是显示分数等电容不等式的非尖锐定量稳定性版本。特别地，我们根据弗伦克尔不对称提供了等电容赤字的下界。此外，我们给出了$ s $-分数容量在$ s $趋于$ 1 $时的渐近性，以及我们的估计相对于参数$ s $的稳定性。
{"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}
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.
{"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}
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