Pub Date : 2023-12-05DOI: 10.1007/s10884-023-10330-x
Hélène Eynard-Bontemps, Andrés Navas
We investigate conjugacy classes of germs of hyperbolic 1-dimensional vector fields at the origin in low regularity. We show that the classical linearization theorem of Sternberg strongly fails in this setting by providing explicit uncountable families of mutually non-conjugate flows with the same multipliers, where conjugacy is considered in the bi-Lipschitz, (C^1) and (C^{1+ac}) settings.
{"title":"On the Failure of Linearization for Germs of $$C^1$$ Hyperbolic Vector Fields in Dimension One","authors":"Hélène Eynard-Bontemps, Andrés Navas","doi":"10.1007/s10884-023-10330-x","DOIUrl":"https://doi.org/10.1007/s10884-023-10330-x","url":null,"abstract":"<p>We investigate conjugacy classes of germs of hyperbolic 1-dimensional vector fields at the origin in low regularity. We show that the classical linearization theorem of Sternberg strongly fails in this setting by providing explicit uncountable families of mutually non-conjugate flows with the same multipliers, where conjugacy is considered in the bi-Lipschitz, <span>(C^1)</span> and <span>(C^{1+ac})</span> settings.\u0000</p>","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"173 3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547649","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}
Pub Date : 2023-11-27DOI: 10.1007/s10884-023-10329-4
Yohei Yamazaki
In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov–Kuznetsov equation on ({mathbb {R}} times {mathbb {T}}_L) and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag (SIAM J Math Anal 44:1175–1210, 2012). Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod (Ann Inst H Poincaré Anal Non Lineaire 32:347–371, 2015) and modifying the mobile distance in Nakanishi and Schlag (2012), we construct a contraction map on the graph space.
本文对({mathbb {R}} times {mathbb {T}}_L)上的Zakharov-Kuznetsov方程构造了不稳定线孤立波的中心稳定流形，并给出了不稳定线孤立波在中心稳定流形上的轨道稳定性，得到了不稳定线孤立波在中心稳定流形附近的渐近稳定性。该构造基于Nakanishi和Schlag (SIAM J Math Anal 44:11 175 - 1210, 2012)的图变换方法。应用Molinet和Pilod (Ann Inst H poincar Anal nonlineaire 32:34 47 - 371, 2015)对傅里叶限制空间的双线性估计，并修改Nakanishi和Schlag(2012)的移动距离，我们在图空间上构造了一个收缩映射。
{"title":"Center Stable Manifolds Around Line Solitary Waves of the Zakharov–Kuznetsov Equation","authors":"Yohei Yamazaki","doi":"10.1007/s10884-023-10329-4","DOIUrl":"https://doi.org/10.1007/s10884-023-10329-4","url":null,"abstract":"<p>In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov–Kuznetsov equation on <span>({mathbb {R}} times {mathbb {T}}_L)</span> and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag (SIAM J Math Anal 44:1175–1210, 2012). Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod (Ann Inst H Poincaré Anal Non Lineaire 32:347–371, 2015) and modifying the mobile distance in Nakanishi and Schlag (2012), we construct a contraction map on the graph space.\u0000</p>","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"66 4","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138524967","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}
Pub Date : 2023-11-06DOI: 10.1007/s10884-023-10326-7
Francesco Fanelli, Rafael Granero-Belinchón
We study the Kolomogorov two-equation model of turbulence in one space dimension. Two are the main results of the paper. First of all, we establish a local well-posedness theory in Sobolev spaces even in the case of vanishing mean turbulent kinetic energy. Then, we show that there are smooth solutions which blow up in finite time. To the best of our knowledge, these results are the first establishing the well-posedness of the system for vanishing initial data and the occurence of finite time singularities for the model under study.
{"title":"Well-Posedness and Singularity Formation for the Kolmogorov Two-Equation Model of Turbulence in 1-D","authors":"Francesco Fanelli, Rafael Granero-Belinchón","doi":"10.1007/s10884-023-10326-7","DOIUrl":"https://doi.org/10.1007/s10884-023-10326-7","url":null,"abstract":"We study the Kolomogorov two-equation model of turbulence in one space dimension. Two are the main results of the paper. First of all, we establish a local well-posedness theory in Sobolev spaces even in the case of vanishing mean turbulent kinetic energy. Then, we show that there are smooth solutions which blow up in finite time. To the best of our knowledge, these results are the first establishing the well-posedness of the system for vanishing initial data and the occurence of finite time singularities for the model under study.","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"23 49","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135545535","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}
Pub Date : 2023-10-28DOI: 10.1007/s10884-023-10325-8
Irena Lasiecka, José H. Rodrigues
{"title":"On weak/Strong Attractor for a 3-D Structural-Acoustic Interaction with Kirchhoff–Boussinesq Elastic Wall Subject to Restricted Boundary Dissipation","authors":"Irena Lasiecka, José H. Rodrigues","doi":"10.1007/s10884-023-10325-8","DOIUrl":"https://doi.org/10.1007/s10884-023-10325-8","url":null,"abstract":"","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136159439","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}
Pub Date : 2023-10-27DOI: 10.1007/s10884-023-10324-9
Changfeng Liu, Shangjiang Guo
{"title":"Boundedness and Stabilization in a Stage-Structured Predator–Prey Model with Two Taxis Mechanisms","authors":"Changfeng Liu, Shangjiang Guo","doi":"10.1007/s10884-023-10324-9","DOIUrl":"https://doi.org/10.1007/s10884-023-10324-9","url":null,"abstract":"","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"20 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234518","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}
Pub Date : 2023-10-27DOI: 10.1007/s10884-023-10302-1
Xiankun Ren, Xueting Tian, Yunhua Zhou
{"title":"On the Topological Entropy of Saturated Sets for Amenable Group Actions","authors":"Xiankun Ren, Xueting Tian, Yunhua Zhou","doi":"10.1007/s10884-023-10302-1","DOIUrl":"https://doi.org/10.1007/s10884-023-10302-1","url":null,"abstract":"","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136262829","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}
Pub Date : 2023-10-21DOI: 10.1007/s10884-023-10323-w
Tran Dinh Ke, Nguyen Nhu Thang
{"title":"An Optimal Halanay Inequality and Decay Rate of Solutions to Some Classes of Nonlocal Functional Differential Equations","authors":"Tran Dinh Ke, Nguyen Nhu Thang","doi":"10.1007/s10884-023-10323-w","DOIUrl":"https://doi.org/10.1007/s10884-023-10323-w","url":null,"abstract":"","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"48 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513170","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}
Pub Date : 2023-10-20DOI: 10.1007/s10884-023-10315-w
Luís Barreira, Claudia Valls
{"title":"Nonautonomous Normal Forms with Parameters","authors":"Luís Barreira, Claudia Valls","doi":"10.1007/s10884-023-10315-w","DOIUrl":"https://doi.org/10.1007/s10884-023-10315-w","url":null,"abstract":"","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617779","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}
Pub Date : 2023-10-20DOI: 10.1007/s10884-023-10322-x
Xiangming Zhu, Chengkui Zhong
{"title":"A Note on the Polynomially Attracting Sets for Dynamical Systems","authors":"Xiangming Zhu, Chengkui Zhong","doi":"10.1007/s10884-023-10322-x","DOIUrl":"https://doi.org/10.1007/s10884-023-10322-x","url":null,"abstract":"","PeriodicalId":15624,"journal":{"name":"Journal of Dynamics and Differential Equations","volume":"9 32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135567054","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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