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On the solutions of nonlocal 1-Laplacian equation with $ L^1 $-data 关于有 $ L^1 $ 数据的非局部 1-拉普拉斯方程的解
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-11-01 DOI: 10.3934/dcds.2023148
Dingding Li, Chao Zhang
{"title":"On the solutions of nonlocal 1-Laplacian equation with $ L^1 $-data","authors":"Dingding Li, Chao Zhang","doi":"10.3934/dcds.2023148","DOIUrl":"https://doi.org/10.3934/dcds.2023148","url":null,"abstract":"We study the solutions to a nonlocal 1-Laplacian equation given by $$ 2text{P.V.}int_{mathbb{R}^N}frac{u(x)-u(y)}{|u(x)-u(y)|} frac{dy}{|x-y|^{N+s}}=f(x) quad textmd{for } xin Omega, $$ with Dirichlet boundary condition $u(x)=0$ in $mathbb R^Nbackslash Omega$ and nonnegative $L^1$-data. By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^+$, we introduce a suitable definition of solutions and prove that the limit function $u$ of ${u_p}$ is a solution of the nonlocal $1$-Laplacian equation above.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139295676","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}
引用次数: 0
Transmission of fast solitons for the NLS with an external potential 具有外部势能的 NLS 的快速孤子传输
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-10-10 DOI: 10.3934/dcds.2023142
Christopher C. Hogan, Jason Murphy
{"title":"Transmission of fast solitons for the NLS with an external potential","authors":"Christopher C. Hogan, Jason Murphy","doi":"10.3934/dcds.2023142","DOIUrl":"https://doi.org/10.3934/dcds.2023142","url":null,"abstract":"We consider the dynamics of a boosted soliton evolving under the cubic NLS with an external potential. We show that for sufficiently large velocities, the soliton is effectively transmitted through the potential. This result extends work of Holmer, Marzuola, and Zworski, who considered the case of a delta potential with no bound states, and work of Datchev and Holmer, who considered the case of a delta potential with a linear bound state.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"102 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139321036","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}
引用次数: 0
On regularity of conjugacy between linear cocycles over partially hyperbolic systems 论部分双曲系统上线性环之间共轭的正则性
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-09-17 DOI: 10.3934/dcds.2023145
B. Kalinin, V. Sadovskaya
{"title":"On regularity of conjugacy between linear cocycles over partially hyperbolic systems","authors":"B. Kalinin, V. Sadovskaya","doi":"10.3934/dcds.2023145","DOIUrl":"https://doi.org/10.3934/dcds.2023145","url":null,"abstract":"We consider H\"older continuous $GL(d,mathbb R)$-valued cocycles, and more generally linear cocycles, over an accessible volume-preserving center-bunched partially hyperbolic diffeomorphism. We study the regularity of a conjugacy between two cocycles. We establish continuity of a measurable conjugacy between {em any} constant $GL(d,mathbb R)$-valued cocycle and its perturbation. We deduce this from our main technical result on continuity of a measurable conjugacy between a fiber bunched linear cocycle and a cocycle with a certain block-triangular structure. The latter class covers constant cocycles with one Lyapunov exponent. We also establish a result of independent interest on continuity of measurable solutions for twisted vector-valued cohomological equations over partially hyperbolic systems. In addition, we give more general versions of earlier results on regularity of invariant subbudles, Riemannian metrics, and conformal structures.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"56 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139339396","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}
引用次数: 0
Characterizations of distality via weak equicontinuity 用弱等连续性描述距离
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-08-31 DOI: 10.3934/dcds.2023096
Jian Li, Y. Yang
{"title":"Characterizations of distality via weak equicontinuity","authors":"Jian Li, Y. Yang","doi":"10.3934/dcds.2023096","DOIUrl":"https://doi.org/10.3934/dcds.2023096","url":null,"abstract":"For an infinite discrete group $G$ acting on a compact metric space $X$, we introduce several weak versions of equicontinuity along subsets of $G$ and show that if a minimal system $(X,G)$ admits an invariant measure then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous; if the product system $(Xtimes X,G)$ of a minimal system $(X,G)$ has a dense set of minimal points, then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous if and only if it is pairwise central$^*$-equicontinuous; if $(X,G)$ is a minimal system with $G$ being abelian, then $(X,G)$ is a system of order $infty$ if and only if it is pairwise FIP$^*$-equicontinuous.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"16 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81816682","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}
引用次数: 0
Failure of Khintchine-type results along the polynomial image of IP0 sets 沿 IP0 集多项式图像的 Khintchine 型结果的失败
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-07-17 DOI: 10.3934/dcds.2023152
Rigoberto Zelada
{"title":"Failure of Khintchine-type results along the polynomial image of IP0 sets","authors":"Rigoberto Zelada","doi":"10.3934/dcds.2023152","DOIUrl":"https://doi.org/10.3934/dcds.2023152","url":null,"abstract":"In\"IP-sets and polynomial recurrence\", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,mathcal A,mu,T)$, any non-constant polynomial $pinmathbb Z[x]$ with $p(0)=0$, any $Ainmathcal A$, and any $epsilon>0$, the set $$R_epsilon^p(A)={ninmathbb N,|,mu(Acap T^{-p(n)}A)>mu^2(A)-epsilon}$$ is IP$^*$, meaning that for any increasing sequence $(n_k)_{kinmathbb N}$ in $mathbb N$, $$text{FS}((n_k)_{kinmathbb N})cap R_epsilon^p(A)neq emptyset,$$ where $$text{FS}((n_k)_{kinmathbb N})={sum_{jin F}n_j,|,Fsubseteq mathbb N,text{ is finite}text{ and }Fneqemptyset}={n_{k_1}+cdots+n_{k_t},|,k_1<cdots<k_t,,tinmathbb N}.$$ In view of the potential new applications to combinatorics, this result has led to the question of whether a further strengthening of Khintchine's recurrence theorem holds, namely whether the set $R_epsilon^p(A)$ is IP$_0^*$ meaning that there exists a $tinmathbb N$ such that for any finite sequence $n_1<cdots<n_t$ in $mathbb N$, $${sum_{jin F}n_j,|,Fsubseteq {1,...,t}text{ and }Fneq emptyset}cap R_epsilon^p(A)neq emptyset.$$ In this paper we give a negative answer to this question by showing that for any given polynomial $pinmathbb Z[x]$ with deg$(p)>1$ and $p(0)=0$ there is an invertible probability preserving system $(X,mathcal A,mu,T)$, a set $Ainmathcal A$, and an $epsilon>0$ for which the set $R_epsilon^p(A)$ is not IP$_0^*$.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"85 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139358733","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}
引用次数: 0
Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise 具有乘法噪声的二维纳维-斯托克斯方程不变度量的唯一性和渐近稳定性
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-07-07 DOI: 10.3934/dcds.2023102
B. Ferrario, M. Zanella
{"title":"Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise","authors":"B. Ferrario, M. Zanella","doi":"10.3934/dcds.2023102","DOIUrl":"https://doi.org/10.3934/dcds.2023102","url":null,"abstract":"We establish the uniqueness and the asymptotic stability of the invariant measure for the two dimensional Navier Stokes equations driven by a multiplicative noise which is either bounded or with a sublinear or a linear growth. We work on an effectively elliptic setting, that is we require that the range of the covariance operator contains the unstable directions. We exploit the generalized asymptotic coupling techniques of Glatt Holtz,Mattingly,Richards(2017) and Kulik,Scheutzow(2018), used by these authors for the stochastic Navier Stokes equations with additive noise. Here we show how these methods are flexible enough to deal with multiplicative noise as well. A crucial role in our argument is played by the Foias Prodi estimate in expected valued, which has a different form (exponential or polynomial decay) according to the growth condition of the multiplicative noise.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"21 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139361803","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}
引用次数: 0
Stress blow-up analysis when suspending rigid particles approach boundary in 3D Stokes flow 三维斯托克斯流中悬浮刚性粒子接近边界时的应力膨胀分析
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-06-06 DOI: 10.3934/dcds.2023141
Haigang Li, Longjuan Xu, Peihao Zhang
{"title":"Stress blow-up analysis when suspending rigid particles approach boundary in 3D Stokes flow","authors":"Haigang Li, Longjuan Xu, Peihao Zhang","doi":"10.3934/dcds.2023141","DOIUrl":"https://doi.org/10.3934/dcds.2023141","url":null,"abstract":"The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"33 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139370729","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}
引用次数: 0
On a structure of non-wandering set of an $ Omega $-stable 3-diffeomorphism possessing a hyperbolic attractor 具有双曲吸引子的稳定3-微分同胚的非游走集结构
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-05-31 DOI: 10.3934/dcds.2023094
M. Barinova, O. Pochinka, E. Yakovlev
{"title":"On a structure of non-wandering set of an $ Omega $-stable 3-diffeomorphism possessing a hyperbolic attractor","authors":"M. Barinova, O. Pochinka, E. Yakovlev","doi":"10.3934/dcds.2023094","DOIUrl":"https://doi.org/10.3934/dcds.2023094","url":null,"abstract":"This paper belongs to a series of papers devoted to the study of the structure of the non-wandering set of an A-diffeomorphism. We study such set $NW(f)$ for an $Omega$-stable diffeomorphism $f$, given on a closed connected 3-manifold $M^3$. Namely, we prove that if all basic sets in $NW(f)$ are trivial except attractors, then every non-trivial attractor is either one-dimensional non-orientable or two-dimensional expanding.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86472323","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}
引用次数: 0
Wellposedness of an elliptic-dispersive coupled system for MEMS MEMS椭圆色散耦合系统的适位性
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-04-27 DOI: 10.3934/dcds.2023055
H. Gimperlein, Runan He, A. Lacey
{"title":"Wellposedness of an elliptic-dispersive coupled system for MEMS","authors":"H. Gimperlein, Runan He, A. Lacey","doi":"10.3934/dcds.2023055","DOIUrl":"https://doi.org/10.3934/dcds.2023055","url":null,"abstract":"In this work, we study the local wellposedness of the solution to a nonlinear elliptic-dispersive coupled system which serves as a model for a Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device consists of two parallel plates separated by a gas-filled thin gap. The nonlinear elliptic-dispersive coupled system modelling the device combines a linear elliptic equation for the gas pressure with a semilinear dispersive equation for the gap width. We show the local-in-time existence of strict solutions for the system, by combining elliptic regularity results for the elliptic equation, Lipschitz continuous dependence of its solution on that of the dispersive equation, and then local-in-time existence for a resulting abstract dispersive problem. Semigroup approaches are key to solve the abstract dispersive problem.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80819241","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}
引用次数: 1
Non-injectivity of infinite interval exchange transformations and generalized Thue-Morse sequences 无限区间交换变换的非注入性与广义Thue-Morse序列
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-04-19 DOI: 10.3934/dcds.2023048
Luis-Miguel Lopez, Philippe Narbel
{"title":"Non-injectivity of infinite interval exchange transformations and generalized Thue-Morse sequences","authors":"Luis-Miguel Lopez, Philippe Narbel","doi":"10.3934/dcds.2023048","DOIUrl":"https://doi.org/10.3934/dcds.2023048","url":null,"abstract":"In this paper we study the non-injectivity arising in infinite interval exchange transformations. In particular, we build and analyze an infinite family of infinite interval exchanges semi-conjugated to generalized Thue-Morse subshifts, whose non-injectivity occurs at a characterizable finite set of points.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73247303","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}
引用次数: 0
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