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

Michele Coti Zelati; Piotr Kalita

doi:10.3934/dcdsb.2017110

Article Contents

2017, Volume 22, Issue 5: 1857-1873. Doi: 10.3934/dcdsb.2017110

This issue Previous Article Strong trajectory and global

$\mathbf{W^{1,p}}$

-attractors for the damped-driven Euler system in

$\mathbb R^2$

Next Article Global attractors of impulsive parabolic inclusions

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

Michele Coti Zelati^1, , and
Piotr Kalita^2,

1.
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
2.
Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland

* Corresponding author: Michele Coti Zelati
* Corresponding author: Michele Coti Zelati

Received: December 2015

Revised: February 2016

Published: August 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:35Q35, 35B41;Secondary:35B45.

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References

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