Global and exponential attractors for the singularly perturbed extensible beam
Michele Coti Zelati
Discrete & Continuous Dynamical Systems - A 2009, 25(3): 1041-1060 doi: 10.3934/dcds.2009.25.1041
The paper deals with the nonlinear evolution equation

ε∂ttu + $\delta$∂tu+ $\omega$∂xxxx u-$[\beta+\int_0^1[\partial_y u(y,t)]^2\d y ]$∂xxu=f,

which describes the motion of the vertical deflection of an extensible Kirchhoff beam. The existence of the global attractor of optimal regularity is shown, as well as the existence of a family of exponential attractors Hölder-continuous in the symmetric Hausdorff distance (with respect to ε) and of finite fractal dimension uniformly bounded with respect to ε.

keywords: exponential attractors. global attractor Extensible beam
Smooth attractors for weak solutions of the SQG equation with critical dissipation
Michele Coti Zelati Piotr Kalita
Discrete & Continuous Dynamical Systems - B 2017, 22(5): 1857-1873 doi: 10.3934/dcdsb.2017110

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.

keywords: Surface quasi-geostrophic equation critical dissipation global attractors Sobolev regularity
Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme
Michele Coti Zelati
Communications on Pure & Applied Analysis 2013, 12(6): 2829-2838 doi: 10.3934/cpaa.2013.12.2829
In this short note, we exploit the tools of multivalued dynamical systems to prove that the stationary statistical properties of the fully implicit Euler scheme converge, as the time-step parameter vanishes, to the stationary statistical properties of the two-dimensional Navier-Stokes equations.
keywords: Multivalued dynamical systems Navier-Stokes equations invariant measures implicit Euler scheme. time discretization

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