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.
Highlights
In recent years a lot of effort has been put to the study of global attractors for problems without uniqueness of solutions
In [23] we have proved, using the single valued theory, that for the problem under consideration there exists the global attractor for the strong solutions, and, using the multivalued theory, that there exists the global attractor for the weak solutions
The new results are contained in Subsection 3.3, where we show that the obtained global attractor has finite fractal dimension
Summary
In recent years a lot of effort has been put to the study of global attractors for problems without uniqueness of solutions. In [23] we proved that weak and strong solutions of problem (1)–(7) given by Definitions 3.1 and 3.2, respectively, exist and that their corresponding attractors coincide. The family of multivalued maps {SH(t)}t≥0 with SH(t) : H → P(H) is given by SH(t)(u0, ω0, θ0) := {(u(t), ω(t), θ(t) : where (u, ω, θ) is (possibly nonunique) weak solution of problem (1)–(7) with the initial data (u0, ω0, θ0) in H, given by Definition 3.1} We define the family of single valued operators {SV (t)}t≥0, SV (t) : V → V, by SV (t)(u0, ω0, θ0) := {(u(t), ω(t), θ(t)) : where (u, ω, θ) is a unique strong solution of problem (1)–(7) with the initial data (u0, ω0, θ0) in V, given by Definition 3.2} We remind the following two Lemmas.
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