Livšic theorem asserts that, for Anosov diffeomorphisms, a Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic orbits are equal to zero. The transfer function is then Lipschitz. We prove a positive Livšic theorem which asserts that a Lipschitz observable is bounded from below by a coboundary if and only if all its Birkhoff sums on periodic orbits are non negative. The new result is that the coboundary can be chosen Lipschitz with a uniform control on the Lipschitz norm. In addition our result holds true for possibly non invertible and not transitive $ C^1 $ maps. We actually prove the main result in the setting of locally maximal hyperbolic sets for general $ C^1 $ map. The construction of the coboundary uses a new notion of the Lax-Oleinik operator that is a standard tool in the discrete Aubry-Mather theory.
{"title":"Lipschitz sub-actions for locally maximal hyperbolic sets of a $ C^1 $ map","authors":"Xifeng Su, Philippe Thieullen, Wenzhe Yu","doi":"10.3934/dcds.2023120","DOIUrl":"https://doi.org/10.3934/dcds.2023120","url":null,"abstract":"Livšic theorem asserts that, for Anosov diffeomorphisms, a Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic orbits are equal to zero. The transfer function is then Lipschitz. We prove a positive Livšic theorem which asserts that a Lipschitz observable is bounded from below by a coboundary if and only if all its Birkhoff sums on periodic orbits are non negative. The new result is that the coboundary can be chosen Lipschitz with a uniform control on the Lipschitz norm. In addition our result holds true for possibly non invertible and not transitive $ C^1 $ maps. We actually prove the main result in the setting of locally maximal hyperbolic sets for general $ C^1 $ map. The construction of the coboundary uses a new notion of the Lax-Oleinik operator that is a standard tool in the discrete Aubry-Mather theory.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135052330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity","authors":"Yinbin Deng, Xin-yan Zhang","doi":"10.3934/dcds.2023093","DOIUrl":"https://doi.org/10.3934/dcds.2023093","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86599066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exponential and sub-exponential stability times for the derivative wave equation","authors":"Yingte Sun, Siming Li, Xiaoqing Wu","doi":"10.3934/dcds.2023007","DOIUrl":"https://doi.org/10.3934/dcds.2023007","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85844352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present note we establish an almost-optimal solvability result for Kirchhoff-type problems of the following form$ begin{cases} -Mbig(|nabla u|^2_{L^2(Omega)}big)Delta u = f(x, u) & text{in } Omega , u geq 0, , unotequiv 0 & text{in } Omega , u = 0 & text{on } partial Omega . end{cases} $where $ f $ has sublinear growth and $ M $ is a non-decreasing map with $ M(0)geq 0 $. Our approach is purely variational, and the result we obtain is resemblant to the one established by Brezis and Oswald (Nonlinear Anal., 1986) for sublinear elliptic equations.
本文建立了以下形式$ begin{cases} -Mbig(|nabla u|^2_{L^2(Omega)}big)Delta u = f(x, u) & text{in } Omega , u geq 0, , unotequiv 0 & text{in } Omega , u = 0 & text{on } partial Omega . end{cases} $的kirchhoff型问题的几乎最优可解性结果，其中$ f $具有次线性增长，$ M $是与$ M(0)geq 0 $的非递减映射。我们的方法是纯变分的，我们得到的结果类似于由Brezis和Oswald(非线性肛门)建立的结果。， 1986)求解次线性椭圆方程。
{"title":"On a Brezis-Oswald-type result for degenerate Kirchhoff problems","authors":"Stefano Biagi, Eugenio Vecchi","doi":"10.3934/dcds.2023122","DOIUrl":"https://doi.org/10.3934/dcds.2023122","url":null,"abstract":"In the present note we establish an almost-optimal solvability result for Kirchhoff-type problems of the following form$ begin{cases} -Mbig(|nabla u|^2_{L^2(Omega)}big)Delta u = f(x, u) & text{in } Omega , u geq 0, , unotequiv 0 & text{in } Omega , u = 0 & text{on } partial Omega . end{cases} $where $ f $ has sublinear growth and $ M $ is a non-decreasing map with $ M(0)geq 0 $. Our approach is purely variational, and the result we obtain is resemblant to the one established by Brezis and Oswald (Nonlinear Anal., 1986) for sublinear elliptic equations.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135212665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by $ ell>0 $ and conserves an $ H^1 $ energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as $ ell $ approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization.This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case.
在本文中，我们提出了标量守恒律的哈密顿正则化，它被参数化为$ ell>0 $，并且守恒$ H^1 $能量。我们证明了这种正则化的全局弱解的存在性。此外，我们证明了当$ well $趋于零时，原始标量守恒律的唯一熵解被恢复，为正则化提供了理由。这种正则化属于一种非扩散、非色散的正则化，最初是为浅水系统开发的，后来扩展到欧拉系统。本文在标量情况下对这类正则化进行了验证。
{"title":"On a Hamiltonian regularization of scalar conservation laws","authors":"Billel Guelmame","doi":"10.3934/dcds.2023118","DOIUrl":"https://doi.org/10.3934/dcds.2023118","url":null,"abstract":"In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by $ ell>0 $ and conserves an $ H^1 $ energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as $ ell $ approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization.This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study strange non-chaotic attractors in a class of quasiperiodically forced monotone interval maps known as pinched skew products. We prove that the probability of positive time-$ N $ Lyapunov exponents—with respect to the unique physical measure on the attractor—decays exponentially as $ Nto infty $. The motivation for this work comes from the study of finite-time Lyapunov exponents as possible early-warning signals of critical transitions in the context of forced dynamics.
我们研究了一类被称为紧缩斜积的拟周期强迫单调区间映射中的奇异非混沌吸引子。我们证明了相对于吸引子上的唯一物理测度的正时间- $ N $ Lyapunov指数的概率呈指数衰减为$ Nto infty $。这项工作的动机来自有限时间李雅普诺夫指数的研究，作为强迫动力学背景下关键转变的可能预警信号。
{"title":"On the probability of positive finite-time Lyapunov exponents on strange nonchaotic attractors","authors":"Flavia Remo, Gabriel Fuhrmann, Tobias Jäger","doi":"10.3934/dcds.2023132","DOIUrl":"https://doi.org/10.3934/dcds.2023132","url":null,"abstract":"We study strange non-chaotic attractors in a class of quasiperiodically forced monotone interval maps known as pinched skew products. We prove that the probability of positive time-$ N $ Lyapunov exponents—with respect to the unique physical measure on the attractor—decays exponentially as $ Nto infty $. The motivation for this work comes from the study of finite-time Lyapunov exponents as possible early-warning signals of critical transitions in the context of forced dynamics.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135505958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rodrigo Arruda, Bernardo Carvalho, Alberto Sarmiento
This paper discusses the dynamics of continuum-wise hyperbolic surface homeomorphisms. We prove that $cw_F$-hyperbolic surface homeomorphisms containing only a finite set of spines are $cw_2$-hyperbolic. In the case of $cw_3$-hyperbolic homeomorphisms we prove the finiteness of spines and, hence, that $cw_3$-hyperbolicity implies $cw_2$-hyperbolicity. In the proof, we adapt techniques of Hiraide [11] and Lewowicz [15] in the case of expansive surface homeomorphisms to prove that local stable/unstable continua of $cw_F$-hyperbolic homeomorphisms are continuous arcs. We also adapt techniques of Artigue, Pac'ifico and Vieitez [6] in the case of N-expansive surface homeomorphisms to prove that the existence of spines is strongly related to the existence of bi-asymptotic sectors and conclude that spines are necessarily isolated from other spines.
{"title":"Continuum-wise hyperbolic homeomorphisms on surfaces","authors":"Rodrigo Arruda, Bernardo Carvalho, Alberto Sarmiento","doi":"10.3934/dcds.2023125","DOIUrl":"https://doi.org/10.3934/dcds.2023125","url":null,"abstract":"This paper discusses the dynamics of continuum-wise hyperbolic surface homeomorphisms. We prove that $cw_F$-hyperbolic surface homeomorphisms containing only a finite set of spines are $cw_2$-hyperbolic. In the case of $cw_3$-hyperbolic homeomorphisms we prove the finiteness of spines and, hence, that $cw_3$-hyperbolicity implies $cw_2$-hyperbolicity. In the proof, we adapt techniques of Hiraide [11] and Lewowicz [15] in the case of expansive surface homeomorphisms to prove that local stable/unstable continua of $cw_F$-hyperbolic homeomorphisms are continuous arcs. We also adapt techniques of Artigue, Pac'ifico and Vieitez [6] in the case of N-expansive surface homeomorphisms to prove that the existence of spines is strongly related to the existence of bi-asymptotic sectors and conclude that spines are necessarily isolated from other spines.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134884812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The separation property for 2D Cahn-Hilliard equations: Local, nonlocal and fractional energy cases","authors":"C. Gal, A. Giorgini, M. Grasselli","doi":"10.3934/dcds.2023010","DOIUrl":"https://doi.org/10.3934/dcds.2023010","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80401640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Propagation dynamics of a nonlocal reaction-diffusion system","authors":"B. Han, D. Kong","doi":"10.3934/dcds.2023028","DOIUrl":"https://doi.org/10.3934/dcds.2023028","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72379345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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