Weak global attractor for the 3D3D-Navier-Stokes equations via the globally modified Navier-Stokes equations

In this paper we obtain the existence of a weak global attractor for the three-dimensional Navier-Stokes equations, that is, a weakly compact set with an invariance property, that uniformly attracts solutions, with respect to the weak topology, for initial data in bounded sets. To that end, we define this weak global attractor in terms of limits of solutions of the globally modified Navier-Stokes equations in the weak topology. We use the theory of semilinear parabolic equations and $\epsilon$-regularity to obtain the local well posedness for the globally modified Navier-Stokes equations and the existence of a global attractor and its regularity.Comment: 17 page

Impulsive surfaces on dynamical systems

This work is devoted to the construction of impulsive sets in Rn. In the literature, there are many examples of impulsive dynamical systems whose impulsive sets are chosen in an abstract way, and in this paper we present sufficient conditions to characterize impulsive sets in Rn which satisfy some “tube conditions” and ensure a good behavior of the flow. Moreover, we present some examples to illustrate the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Fundação de Amparo à Pesquisa do Estado de São Paul

Attractors for impulsive non-autonomous dynamical systems and their relations

In this work, we deal with several different notions of attractors that may appear in the impulsive non-autonomous case and we explore their relationships to obtain properties regarding the different scenarios of asymptotic dynamics, such as the cocycle attractor, the uniform attractor and the global attractor for the impulsive skew-product semiflow. Lastly, we illustrate our theory by exhibiting an example of a non-classical non-autonomous parabolic equation with subcritical nonlinearity and impulses.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Fundação de Amparo à Pesquisa do Estado de São Paul

Skew Product Semiflows and Morse Decomposition

This paper is devoted to the investigation of the dynamics of non-autonomous differential equations. The description of the asymptotic dynamics of non-autonomous equations lies on dynamical structures of some associated limiting non-autonomous - and autonomous - differential equations (one for each global solution in the attractor of the driving semigroup of the associated skew product semi-flow). In some cases, we have infinitely many limiting problems (in contrast with the autonomous - or asymptotically autonomous - case for which we have only one limiting problem; that is, the semigroup itself). We concentrate our attention in the study of the Morse decomposition of attractors for these non-autonomous limiting problems as a mean to understand some of the asymptotics of our non-autonomous differential equations. In particular, we derive a Morse decomposition for the global attractors of skew product semiflows (and thus for pullback attractors of non-autonomous differential equations) from a Morse decomposition of the attractor for the associated driving semigroup. Our theory is well suited to describe the asymptotic dynamics of non-autonomous differential equations defined on the whole line or just for positive times, or for differential equations driven by a general semigroup

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

Estrutura de atratores e estimativas de suas dimensões fractais

This work is dedicated to the study of the structure of attractors of dynamical systems with the objective of estimating their fractal dimension. First we study the case of exponential global attractors of some generalized gradient-like semigroups in a general Banach space, and estimate their fractal dimension in terms of themaximumof the dimension of the local unstablemanifolds of the isolated invariant sets, Lipschitz properties of the semigroup and rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A, \'A POT. \') is an attractor-repeller pair for the attractor A of a semigroup {T (t ) : t 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of \'A POT. \', the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. Also, making use of the skew product semiflow and its Morse decomposition, we give some estimates of the fractal dimension of the pullback attractors of non-autonomous dynamical systemsEste trabalho é dedicado ao estudo da estrutura dos atratores de sistemas dinâmicos com o objetivo de obter estimativas de suas dimensões fractais. Primeiramente estudamos o caso de atratores globais exponenciais de alguns semigrupos gradient-like generalizados em um espaço de Banach geral, e estimamos suas dimensões fractais em termos da máxima dimensão das variedades instáveis locais dos conjuntos invariantes isolados, a propriedades de Lipschitz do semigrupo e da taxa de atração exponencial. Também generalizamos este resultado para alguns processos de evoluções especiais, introduzindo um conceito de decomposição de Morse com atração pullback. Sob hipóteses apropriadas, se (A, \'A POT. \') é um par atrator-repulsor para o atratorA de um semigrupo {T (t ) : t 0}, então a dimensão fractal de A pode ser estimada em termos da dimensão fractal da variedade instável de \'A POT. \', a dimensão fractal de A, as propriedades de Lipschitz do semigrupo e a taxa de atração exponencial. Os ingredientes da demonstração são a noção de semigrupos gradient-like e seus atratores regulares, decomposição de Morse e uma análise fina da estrutura dos atratores. Além disto, fazendo uso do skew product semiflow e sua decomposição de Morse, damos estimativas da dimensão fractal dos atratores pullback de sistêmas dinâmicos não-autônomo

Discrete dynamical systems attractors: fractal dimension and continuity of the structure under perturbations

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractaisIn this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractor

Structure of attractors for skew product semiflows

In this work we study the continuity and structural stability of the uniform attractor associated with non-autonomous perturbations of differential equations. By a careful study of the different definitions of attractor in the non-autonomous framework, we introduce the notion of lifted-invariance on the uniform attractor, which becomes compatible with the dynamics in the global attractor of the associated skew product semiflow, and allows us to describe the internal dynamics and the characterization of the uniform attractors. The associated pullback attractors and their structural stability under perturbations will play a crucial role.FAPESP (grant 2012/23724-1, 2008/55516-3)CNPq (grant 305230/2011-5)CAPES/DGU (grant 238/2011)FEDER (European Union) and Ministerio de Economía y Competitividad (grant # MTM2011-22411)Proyecto de Excelencia (grant FQM-1492

Upper and Lower Semicontinuity of Impulsive Cocycle Attractors for Impulsive Nonautonomous Systems

In this work we present results to ensure a weak upper semicontinuity for a family of impulsive cocycle attractors of nonautonomous impulsive dynamical systems, as well as an example of nonautonomous dynamical system generated by an ODE in the real line to illustrate our results. Moreover, we present theoretical results regarding lower semicontinuity of impulsive cocycle attractors

Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems

This is the accepted version of the following article: Everaldo de Mello Bonotto, Matheus Cheque Bortolan, Rodolfo Collegari, José Manuel Uzal. Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems. Discrete and Continuous Dynamical Systems - B, 2021, 26(9): 4645-4661. doi: 10.3934/dcdsb.2020306In this paper we investigate the long time behavior of a nonautonomous dynamical system (cocycle) when its driving semigroup is subjected to impulses. We provide conditions to ensure the existence of global attractors for the associated impulsive skew-product semigroups, uniform attractors for the coupled impulsive cocycle and pullback attractors for the associated evolution processes. Finally, we illustrate the theory with an application to cascade systems.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu