On the Theory of Global Attractors and Lyapunov Functionals

Zelati, Michele Coti

doi:10.1007/s11228-012-0215-2

Cited by 19 publications

(9 citation statements)

References 28 publications

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“…THEOREM A.4 (cf. [18,Theorem 4.6], [19,Proposition 4.2]). If the multivalued semiflow {S(t)} t∈R + is t * -closed and possesses a compact absorbing set, then it has a global attractor A.…”

Section: Appendix a Multivalued Semiflowsmentioning

confidence: 99%

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Smooth attractors for weak solutions of the SQG equation with critical dissipation

Zelati¹,

Kalita²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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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.2000 Mathematics Subject Classification. 35Q35, 35B41, 35B45.

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Section: Appendix a Multivalued Semiflowsmentioning

confidence: 99%

“…The compactness requirement for the absorbing set is certainly not needed in general, as more appropriate notions of asymptotic compactness can be shown to be equivalent to the existence of the global attractor. The reader is referred to [18,19,24,29] for more details.…”

Section: Appendix a Multivalued Semiflowsmentioning

confidence: 99%

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Zelati¹,

Kalita²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…These results further justify the use of the NavierStokes-Voigt equations as an inviscid regularization of the 3D NSE, in particular for numerical computations and simulations. For the final issue (c), based on the concept of multivalued semiflows [10,38], we derive results on the convergence of the (strong) global attractors for the Voigt model to the (weak) global attractor for the 3D NSE, as α goes to zero. We also derive conditions for the weak global attractor of the NSE to be strong, in terms of the topologies of the above mentioned convergences.…”

Section: Introductionmentioning

confidence: 99%

Singular Limits of Voigt Models in Fluid Dynamics

Zelati

Gal

2015

J. Math. Fluid Mech.

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ABSTRACT. We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi in [33].

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“…The fact that viscosity solutions are strongly continuous is a consequence of the fact that they satisfy the energy equality (see [3,7] for a proof in the critical case). Following the approach in [13,27], for t ≥ 0 and each θ 0 ∈ L 2 we define the set-valued maps S γ (t) : L 2 → 2 L 2 , still denoted as the single-valued ones,…”

Section: Global Attractors For Weak Solutionsmentioning

confidence: 99%

Long-Time Behavior and Critical Limit of Subcritical SQG Equations in Scale-Invariant Sobolev Spaces

Zelati

2017

J Nonlinear Sci

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We consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the L ∞ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attractors and prove their stability and upper-semicontinuity with respect to the strength of the diffusion. THEOREM 1.1. Let γ ∈ (1, 2) be fixed, and assume that f ∈ L ∞ ∩ H 2−γ . The dynamical system S γ (t) generated by (SQG γ ) on H 2−γ possesses a unique invariant global attractor A γ , bounded in H 2−γ/2 , and therefore compact in H 2−γ . In particular,Another important feature of the attractors A γ is their stability with respect to the parameter γ, as γ → 1 + . Namely, the following uniform estimates and upper semicontinuity result hold.2000 Mathematics Subject Classification. 35Q35, 35B41, 35B45.

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On the Theory of Global Attractors and Lyapunov Functionals

Cited by 19 publications

References 28 publications

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Singular Limits of Voigt Models in Fluid Dynamics

Long-Time Behavior and Critical Limit of Subcritical SQG Equations in Scale-Invariant Sobolev Spaces

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