Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications

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

“…The fundamental hypotheses on the concrete evolutionary equations is that the trajectory space is a metrizable normal topological space and the natural translation semigroup possesses a compact trajectory attractor. We will investigate these issues in some other papers, see [45–48].…”

“…Here we want to remark that after submission of this manuscript, we also used the trajectory attractor to construct the trajectory statistical solution for the 3D globally modified Navier‐Stokes equations in [45], for the 3D incompressible Navier‐Stokes equations in [47], as well as the strong trajectory statistical solutions for the 2D dissipative Euler equations in [46]. Very recently, Zhao, Li and Caraballo in [48] proved some sufficient conditions ensuring the existence of trajectory statistical solutions for general autonomous evolution equations.…”

Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines

“…Thus, we first prove the existence of a trajectory attractor for such system, which is a minimal compact trajectory attracting set for the natural translation semigroup defined on the trajectory space. Furthermore, based on the abstract results (trajectory attractor approach) developed in [38], we construct trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines.…”

“…Inspired by the notion of generalized Banach limit, they used the weak trajectory attractor to constructed the (weak) trajectory statistical solutions for the 3D globally modified Navier-Stokes equations. Furthermore, they established some abstract results about the trajectory statistical solutions in [38] and applied it to the three dimensional incompressible magneto-micropolar fluids ( [38]), the three dimensional incompressible Navier-Stokes equations ( [39]), nonlinear wave equations ( [18]), dissipative Euler equations ( [40]), Klein-Gordon-Schrödinger equations ( [37]).…”

Random attractors for 2D stochastic micropolar fluid flows on unbounded domains

“…Dong, Li and Wu studied the global regularity and large time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation [22]. More recently, Zhao, Li and Caraballo [45] investigated the trajectory statistical solutions for evolution equations, and the theory developed therein was later applied to study three dimensional incompressible micropolar fluid flows by Zhao, Li, Sang (see [46]).…”