Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

“…We remark that our idea originates from [33,37,38,40]. In [38], Zhao, Li and Caraballo first gave some sufficient conditions on the existence of trajectory statistical solutions for autonomous continuous evolution equations, and then proved that the trajectory statistical solution satisfies a Liouville type equation.…”

“…Based on these works, Zhao and Caraballo [36] used the trajectory attractor to construct trajectory statistical solutions for evolution equations. For more results on this issue of continuous systems, the interested reader is referred to [2,6,4,19,23,25,30,28,29,36,37,39,38,41], etc.…”

“…In [38], Zhao, Li and Caraballo first gave some sufficient conditions on the existence of trajectory statistical solutions for autonomous continuous evolution equations, and then proved that the trajectory statistical solution satisfies a Liouville type equation. Later, Zhao, Caraballo and Lukaszewicz [37] further constructed a statistical solution and showed that the statistical solution satisfies a Liouville type theorem for nonautonomous continuous Klein-Gordon-Schrödinger equations. In this article, we mainly extend the results established in [37] to the case of discrete coupled Schrödinger-Boussinesq equations.…”

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.…”

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]).…”