{"title":"Random Gibbs $ u $-state for partially hyperbolic on fibers system","authors":"Xue Liu","doi":"10.3934/dcds.2023070","DOIUrl":"https://doi.org/10.3934/dcds.2023070","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"179 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73935100","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 work we consider the initial value problem for the generalized Benjamin equation begin{equation}label{Benj-IVP} begin{cases} partial_t u-lmathcal{H} partial_x^2u-partial_x^3u+u^ppartial_xu = 0, quad x,; tin mathbb{R};;;,; pgeq 1, u(x,0) = u_0(x), end{cases} end{equation} where $u=u(x,t)$ is a real valued function, $0
{"title":"Evolution of the radius of analyticity for the generalized Benjamin equation","authors":"Renata O. Figueira, M. Panthee","doi":"10.3934/dcds.2023039","DOIUrl":"https://doi.org/10.3934/dcds.2023039","url":null,"abstract":"In this work we consider the initial value problem for the generalized Benjamin equation begin{equation}label{Benj-IVP} begin{cases} partial_t u-lmathcal{H} partial_x^2u-partial_x^3u+u^ppartial_xu = 0, quad x,; tin mathbb{R};;;,; pgeq 1, u(x,0) = u_0(x), end{cases} end{equation} where $u=u(x,t)$ is a real valued function, $0<l<1$ and $mathcal{H}$ is the Hilbert transform. This model was introduced by T. B. Benjamin (J. Fluid Mech. 245 (1992) 401--411) and describes unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We prove that the local solution to the IVP associated with the generalized Benjamin equation for given data in the spaces of functions analytic on a strip around the real axis continue to be analytic without shrinking the width of the strip in time. We also study the evolution in time of the radius of spatial analyticity and show that it can decrease as the time advances. Finally, we present an algebraic lower bound on the possible rate of decrease in time of the uniform radius of spatial analyticity.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"2 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83783217","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}
The stabilizability to trajectories of the Schl"ogl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optimality conditions are investigated. These conditions are then used to compute stabilizing receding horizon based controls. Results of numerical simulations are presented comparing their stabilizing performance with that of saturated explicit feedback controls.
{"title":"Global stabilizability to trajectories for the Schlögl equation in a Sobolev norm","authors":"K. Kunisch, S. Rodrigues","doi":"10.3934/dcds.2023017","DOIUrl":"https://doi.org/10.3934/dcds.2023017","url":null,"abstract":"The stabilizability to trajectories of the Schl\"ogl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optimality conditions are investigated. These conditions are then used to compute stabilizing receding horizon based controls. Results of numerical simulations are presented comparing their stabilizing performance with that of saturated explicit feedback controls.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"12 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74343132","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 classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ mathbb{R}^4$: begin{equation}left{ begin{aligned}&-Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, quad xin mathbb{R}^4,&(-Delta)^2 v(x) = u^{p_2}(x) e^{q_2v(x)}, quad xin mathbb{R}^4, end{aligned} right. end{equation} where $ 0leq p_1<1$, $ p_2>0$, $ q_1>0$, $ q_2 geq 0$, $ u>0$ and satisfies $$ int_{mathbb{R}^4} u^{p_1}(x) e^{q_1v(x)} dx
{"title":"Classification of solutions for some mixed order elliptic system","authors":"Genggeng Huang, Yating Niu","doi":"10.3934/dcds.2023079","DOIUrl":"https://doi.org/10.3934/dcds.2023079","url":null,"abstract":"In this paper, we classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ mathbb{R}^4$: begin{equation}left{ begin{aligned}&-Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, quad xin mathbb{R}^4,&(-Delta)^2 v(x) = u^{p_2}(x) e^{q_2v(x)}, quad xin mathbb{R}^4, end{aligned} right. end{equation} where $ 0leq p_1<1$, $ p_2>0$, $ q_1>0$, $ q_2 geq 0$, $ u>0$ and satisfies $$ int_{mathbb{R}^4} u^{p_1}(x) e^{q_1v(x)} dx<infty,quad int_{mathbb{R}^4} u^{p_2}(x) e^{q_2 v(x)} dx<infty.$$ Under additional assumptions (H1) or (H2), we study the asymptotic behavior of the solutions to the system and we establish the equivalent integral formula for the system. By using the method of moving spheres, we obtain the classification results of the solutions in the system.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"13 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82358092","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 note we study a new kinetic model of opinion dynamics. The model incorporates two forces -- alignment of opinions under all-to-all communication driving the system to a consensus, and Rayleigh type friction force that drives each `player' to its fixed conviction value. The balance between these forces creates a non-trivial limiting outcome. We establish existence of a global mono-opinion state, whereby any initial distribution of opinions for each conviction value aggregates to the Dirac measure concentrated on a single opinion. We identify several cases where such a state is unique and depends continuously on the initial distribution of convictions. Several regularity properties of the limiting distribution of opinions are presented.
{"title":"Continuous model of opinion dynamics with convictions","authors":"Vinh Nguyen, R. Shvydkoy","doi":"10.3934/dcds.2023076","DOIUrl":"https://doi.org/10.3934/dcds.2023076","url":null,"abstract":"In this note we study a new kinetic model of opinion dynamics. The model incorporates two forces -- alignment of opinions under all-to-all communication driving the system to a consensus, and Rayleigh type friction force that drives each `player' to its fixed conviction value. The balance between these forces creates a non-trivial limiting outcome. We establish existence of a global mono-opinion state, whereby any initial distribution of opinions for each conviction value aggregates to the Dirac measure concentrated on a single opinion. We identify several cases where such a state is unique and depends continuously on the initial distribution of convictions. Several regularity properties of the limiting distribution of opinions are presented.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91162837","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 propose and study a nonlocal conservation law modelling traffic flow in the existence of inter-vehicle communication. It is assumed that the nonlocal information travels at a finite speed and the model involves a space-time nonlocal integral of weighted traffic density. The well-posedness of the model is established under suitable conditions on the model parameters and by a suitably-defined initial condition. In a special case where the weight kernel in the nonlocal integral is an exponential function, the nonlocal model can be reformulated as a $2times2$ hyperbolic system with relaxation. With the help of this relaxation representation, we show that the Lighthill-Whitham-Richards model is recovered in the equilibrium approximation limit.
{"title":"A space-time nonlocal traffic flow model: Relaxation representation and local limit","authors":"Q. Du, Kuang Huang, J. Scott, Wen Shen","doi":"10.3934/dcds.2023054","DOIUrl":"https://doi.org/10.3934/dcds.2023054","url":null,"abstract":"We propose and study a nonlocal conservation law modelling traffic flow in the existence of inter-vehicle communication. It is assumed that the nonlocal information travels at a finite speed and the model involves a space-time nonlocal integral of weighted traffic density. The well-posedness of the model is established under suitable conditions on the model parameters and by a suitably-defined initial condition. In a special case where the weight kernel in the nonlocal integral is an exponential function, the nonlocal model can be reformulated as a $2times2$ hyperbolic system with relaxation. With the help of this relaxation representation, we show that the Lighthill-Whitham-Richards model is recovered in the equilibrium approximation limit.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"33 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79341606","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 show that every codimension one partially hyperbolic diffeomorphism must support on $mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is intrinsically ergodic, and the A. Katok's conjecture about the existence of ergodic measures with intermediate entropies holds for it.
{"title":"On codimension one partially hyperbolic diffeomorphisms","authors":"Xiang Zhang","doi":"10.3934/dcds.2023066","DOIUrl":"https://doi.org/10.3934/dcds.2023066","url":null,"abstract":"We show that every codimension one partially hyperbolic diffeomorphism must support on $mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is intrinsically ergodic, and the A. Katok's conjecture about the existence of ergodic measures with intermediate entropies holds for it.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"49 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78742126","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}
Y. Chitour, S'ebastien Fueyo, Guilherme Mazanti, M. Sigalotti
The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-Lambda_1,dots,t-Lambda_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q in [1, +infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $dmax{Lambda_1,dots,Lambda_N}$, where $d$ is the dimension of the state space.
本文研究有限维线性差分时滞方程的可控性，即给定时间$t$的状态可以用时间$t$的控制值和之前有限多个时间$t-Lambda_1,dots,t-Lambda_N$的状态值的线性组合来表示的动力学。基于yamamoto提出的一般无限维动力系统的实现理论，我们得到了在$L^q$空间，$q in [1, +infty)$中有限时间近似可控的频域充要条件。我们还提供了$L^1$精确可控性的一个必要条件，可以看作是$L^1$近似可控性判据的闭包。进一步，我们提供了近似和精确可控性的最小时间的显式上界，由$dmax{Lambda_1,dots,Lambda_N}$给出，其中$d$是状态空间的维数。
{"title":"Hautus–Yamamoto criteria for approximate and exact controllability of linear difference delay equations","authors":"Y. Chitour, S'ebastien Fueyo, Guilherme Mazanti, M. Sigalotti","doi":"10.3934/dcds.2023049","DOIUrl":"https://doi.org/10.3934/dcds.2023049","url":null,"abstract":"The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-Lambda_1,dots,t-Lambda_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q in [1, +infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $dmax{Lambda_1,dots,Lambda_N}$, where $d$ is the dimension of the state space.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88155682","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}
For a Lipschitz $mathbb{Z}-$periodic function $phi:mathbb{R}to mathbb{R}^2$ satisfied that $mathbb{R}^2setminus{phi(x):xinmathbb{R}}$ is not connected, an integer $bge 2$ and $lambdain (c/{b^{frac12}},1)$, we prove the following for the generalized Weierstrass-type function $W(x)=sumlimits_{n=0}^{infty}{{lambda}^nphi(b^nx)}$: the box dimension of its graph is equal to $3+2log_blambda$, where $c$ is a constant depending on $phi$.
{"title":"Box dimension of the graphs of the generalized Weierstrass-type functions","authors":"Haojie Ren","doi":"10.3934/dcds.2023068","DOIUrl":"https://doi.org/10.3934/dcds.2023068","url":null,"abstract":"For a Lipschitz $mathbb{Z}-$periodic function $phi:mathbb{R}to mathbb{R}^2$ satisfied that $mathbb{R}^2setminus{phi(x):xinmathbb{R}}$ is not connected, an integer $bge 2$ and $lambdain (c/{b^{frac12}},1)$, we prove the following for the generalized Weierstrass-type function $W(x)=sumlimits_{n=0}^{infty}{{lambda}^nphi(b^nx)}$: the box dimension of its graph is equal to $3+2log_blambda$, where $c$ is a constant depending on $phi$.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"50 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79385013","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 introduce a localized version of the nudging data assimilation algorithm for the periodic 2D Navier-Stokes equations in which observations are confined (i.e., localized) to a window that moves across the entire domain along a predetermined path at a given speed. We prove that, if the movement is fast enough, then the algorithm perfectly synchronizes with a reference solution. The analysis suggests an informed scheme in which the subdomain moves according to a region where the error is dominant is optimal. Numerical simulations are presented that compare the efficacy of movement that follows a regular pattern, one guided by the dominant error, and one that is random.
{"title":"Convergence of a mobile data assimilation scheme for the 2D Navier-Stokes equations","authors":"A. Biswas, Z. Bradshaw, M. Jolly","doi":"10.3934/dcds.2023078","DOIUrl":"https://doi.org/10.3934/dcds.2023078","url":null,"abstract":"We introduce a localized version of the nudging data assimilation algorithm for the periodic 2D Navier-Stokes equations in which observations are confined (i.e., localized) to a window that moves across the entire domain along a predetermined path at a given speed. We prove that, if the movement is fast enough, then the algorithm perfectly synchronizes with a reference solution. The analysis suggests an informed scheme in which the subdomain moves according to a region where the error is dominant is optimal. Numerical simulations are presented that compare the efficacy of movement that follows a regular pattern, one guided by the dominant error, and one that is random.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"23 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86060963","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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