The fractal dimension of pullback attractors for the 2D Navier–Stokes equations with delay

Abstract

This paper is concerned with the bounded fractal and Hausdorff dimension of the pullback attractors for 2D nonautonomous incompressible Navier-Stokes equations with constant delay terms. Using the construction of trace formula with two bases for phase spaces of product flow, the upper boundedness of fractal dimension has been achieved.

Numerical modeling of mixed convective nanofluid flow with fractal stochastic heat and mass transfer using finite differences

“…In (Constantin et al, 1985), approximate solutions of NSE in terms of time are provided for fractal dimensions, while in (Hinz and Teplyaev, 2015), NSE on onedimensional topological fractals is studied using Hodge theory. In Comparison of the fractal and fractal stochastic models using α 0.5, β 3, λ 1 1, λ 2 0.7, Pr 1, Rd 0.1, Q 0.1, Nb 0.1, Nt 0.1, Sc 1, γ 0.1 (Yang et al, 2020), the pullback attractors for 2D non-autonomous incompressible NSE with constant delay terms were investigated, and their limited fractal and Hausdorff dimensions were determined. The NSE was also investigated in fractal dimensions of invariant sets (Chepyzhov and Llyin, 2014).…”

Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains

“…Moreover, Garcín-Luengo, Marín-Rubio and Planas [17] studied the 2D Navier-Stokes equations with double-time delays on the convective term and external forces, and obtained the existence of pullback attractors. Additional results can be found in [4], [15], [16], [26], [36], [38] and references therein.…”

Pullback dynamics of a 3D modified Navier-Stokes equations with double delays

“…In addition, the upper semi-continuity of pullback attractor was also studied in [37] when the perturbed external force disappears as parameter tends to zero. The Navier-Stokes equations with delays were firstly considered by Caraballo and Real in [5], then there are a lot of works concerning asymptotic behavior, stability, the existence of pullback attractors and the fractal dimensional of pullback attractors for time-delayed Navier-Stokes equations (see, e.g., [6,7,15,30,38]). It is worth to be pointed out that García-Luengo, Marín-Rubio and Real [15] obtained that the existence of pullback attractors for the 2D Navier-Stokes model with finite delay.…”

“…It is worth to be pointed out that García-Luengo, Marín-Rubio and Real [15] obtained that the existence of pullback attractors for the 2D Navier-Stokes model with finite delay. Furthermore, the bounded fractal and Hausdorff dimension of the pullback attractors for 2D non-autonomous incompressible Navier-Stokes equations with delay was studied in [38]. The above work is to study the time delay which only exists in the external force.…”