Using trajectory attractor to construct trajectory statistical solution for the 3D incompressible micropolar flows

Abstract

In this article, we first prove the existence of trajectory attractor for the threedimensional incompressible micropolar fluids. This trajectory attractor is the minimal compact trajectory attracting set for the natural translation semigroup on the trajectory space. Then we construct the trajectory statistical solution for this micropolar fluids via the natural translation semigroup and trajectory attractor. In our construction the trajectory statistical solution is a space-time probability measure, which is carried by the trajectory attractor and is invariant under the acting of the translation semigroup. K E Y W O R D S invariant measure, micropolar fluids, trajectory attractor, trajectory statistical solution M S C 2 0 1 0 35B41, 34D35, 76F20

Homogenization of Trajectory Statistical Solutions for the 3D Incompressible Micropolar Fluids with Rapidly Oscillating Terms

“…Proceeding as the proofs of [58] (Lemma 2.1), we can obtain the following lemma about the existence and estimate of weak solutions to problem (15) and ( 16). The difference is that we replace the the positive constant R = 2λ…”

“…For every ∈ (0, 1], the existence of the trajectory attractors and trajectory statistical solutions for Equation (15) has been proved in [58]. To be quite explicit, we have the following: Theorem 1 (cf.…”

Statistical solutions for a nonautonomous modified Swift–Hohenberg equation

“…For a much wider class of dissipative semigroups, Chekroun and Glatt-Holtz 30 also applied the generalized Banach limit to constructing the invariant measures, but they generalized and simplified the proofs of Wang and Łukaszewicz et al 28,29 Recently, a series of works developed some techniques to provide a construction of invariant measures for nonautonomous systems with minimal assumptions on the underlying dynamical process (see Foias et al, 27 Wang, 28 and Łukaszewicz et al 29,31,32 ). Nowadays, these theories have been employed to establish the existence of invariant measures and (trajectory) statistical solutions for some evolution equations (see, e.g., other works [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] and the references therein). However, invariant measures in these works were usually discussed in a closed subspace of L 2 (Ω) (see previous studies 27,31,32,45,51,52 ) or its own product space, 47 and their regularity can be considered.…”

Equivalence between invariant measures and statistical solutions for the 2D non‐autonomous magneto‐micropolar fluid equations

“…(1.3) System (1.1) describes the motion of conductive micropolar fluids (see previous works [15][16][17] ) under the action of magnetic fields. The mathematical studies for this kind of system have been widely done due to its important physical background (see, e.g., previous studies [18][19][20][21][22][23][24][25][26][27][28] ).…”