Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations

Abstract

We first prove the existence and regularity of the trajectory attractor for a threedimensional system of globally modified Navier-Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.

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].…”

“…By this way, we avoid the semigroup of solution operators and consequently do not need the uniqueness of the weak solutions. 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.…”

Remarks on nontrivial pullback attractors of the 2‐D Navier‐Stokes equations with delays

“…Let P be the Helmholz‐Leray orthogonal projection in ( L 2 (Ω)) 2 onto the space H . The bilinear operator B (·,·) and trilinear operator b (·,·,·) are defined as (see Foias et al, Temam, Yang et al, Zhao and Caraballo, Zhao and Yang, Zhao et al, and Zhao and Duan) which satisfy in 2‐D …”

Asymptotic Behavior of the Solutions of the Generalized Globally Modified Navier–Stokes Equations

“…Since the modifying factor F N (‖Λ β u‖) decreases the singularity of the quadratic convection term (u • ∇)u, it allows the authors to derive the existence and uniqueness of global solutions [8]. Following [8], the existence results and the asymptotic behaviors of solutions to problem (1) were extensively studied in different contexts, see e.g., [13][14][15][16][17][18][19][20][21] and the review paper [22].…”