The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness
Alexey Cheskidov Songsong Lu
Discrete & Continuous Dynamical Systems - S 2009, 2(1): 55-66 doi: 10.3934/dcdss.2009.2.55
We study the uniform global attractor for a general nonautonomous reaction-diffusion system without uniqueness using a new developed framework of an evolutionary system. We prove the existence and the structure of a weak uniform (with respect to a symbol space) global attractor $\mathcal A$. Moreover, if the external force is normal, we show that this attractor is in fact a strong uniform global attractor. The existence of a uniform (with respect to the initial time) global attractor $\mathcal A^0$ also holds in this case, but its relation to $\mathcal A$ is not yet clear due to the non-uniqueness feature of the system.
keywords: normal symbol. reaction-diffusion system evolutionary system uniform global attractor
Pullback attractors for generalized evolutionary systems
Alexey Cheskidov Landon Kavlie
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 749-779 doi: 10.3934/dcdsb.2015.20.749
We give an abstract framework for studying nonautonomous PDEs, called a generalized evolutionary system. In this setting, we define the notion of a pullback attractor. Moreover, we show that the pullback attractor, in the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our framework to both autonomous and nonautonomous evolutionary systems as they first appeared in earlier works by Cheskidov, Foias, and Lu. In this con- text, we compare the pullback attractor to both the global attractor (in the autonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equations on a periodic domain with a translationally bounded force. We show that the Leray-Hopf weak solutions form a generalized evolutionary system and must then have a weak pullback attractor.
keywords: generalized evolutionary systems Nonautonomous dynamical systems fluid mechanics pullback attractors Navier-Stokes equations.
An inviscid dyadic model of turbulence: The global attractor
Alexey Cheskidov Susan Friedlander Nataša Pavlović
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 781-794 doi: 10.3934/dcds.2010.26.781
Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. In a companion paper [8] it is proved that every solution for the system with forcing blows up in finite time in the Sobolev $H^{5/6}$ norm. In this present paper, it is proved that after the blow-up time all solutions stay in $H^s$, $s < 5/6$ for almost all time. It is proved that the model system exhibits the phenomenon of anomalous (or turbulent) dissipation which was conjectured for the Euler equations by Onsager. As a consequence of this anomalous dissipation the unique equilibrium of the system is a global attractor.
keywords: global attractor turbulence. Dyadic shell model

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