On relation between attractors for single and multivalued semiflows for a certain class of PDEs

Piotr Kalita; Grzegorz Łukaszewicz; Jakub Siemianowski

doi:10.3934/dcdsb.2019012

Article Contents

2019, Volume 24, Issue 3: 1199-1227. Doi: 10.3934/dcdsb.2019012

This issue Previous Article Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations Next Article Attractors of multivalued semi-flows generated by solutions of optimal control problems

On relation between attractors for single and multivalued semiflows for a certain class of PDEs

1.
Faculty of Mathematics and Computer Sciences, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
2.
Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
3.
Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Received: December 31, 2017

Revised: April 30, 2018

Early access: January 2019

Published: January 2019

Work of P.K.G.L,and J.S.was supported by National Science Center (NCN) of Poland under project No.DEC-2017/25/B/ST1/00302,work of P.K.and J.S.was partially supported by NCN of Poland under project No.UMO-2016/22/A/ST1/00077

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Abstract

Sometimes it is not possible to prove the uniqueness of the weak solutions for problems of mathematical physics, but it is possible to bootstrap their regularity to the regularity of strong solutions which are unique. In this paper we formulate an abstract setting for such class of problems and we provide the conditions under which the global attractors for both strong and weak solutions coincide and the fractal dimension of the common attractor is finite. We present two problems belonging to this class: planar Rayleigh–Bénard flow of thermomicropolar fluid and surface quasigeostrophic equation on torus.

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Mathematics Subject Classification: Primary: 35B41, 35B65, 76D03, 80M35.

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