DCDS
Global and exponential attractors for the singularly perturbed extensible beam
Michele Coti Zelati
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
DCDS-B
Smooth attractors for weak solutions of the SQG equation with critical dissipation
Michele Coti Zelati Piotr Kalita

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
CPAA
Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme
Michele Coti Zelati
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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