Dynamics of a globally modified Navier–Stokes model with double delay

Abstract

This paper investigates the dynamics of a class of three-dimensional globally modified Navier-Stokes equations with double delay in the forcing and convective terms. We first prove the well-posedness of solutions of such system, which enables us to establish suitable non-autonomous dynamical systems. We then show the existence and uniqueness of pullback attractors for the associated dynamical systems. Finally, by using the generalized Banach limit, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors.

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

“…After that, the existence and asymptotic behaviors of solutions of 3D GMNSE have been investigated in Refs. 4–7. And the invariant measures as well as the Liouville‐type theorem of 3D GMNSE have been examined in Refs.…”

“…And the invariant measures as well as the Liouville‐type theorem of 3D GMNSE have been examined in Refs. 7, 8 and the references therein.…”

Stochastic 3D Globally Modified Navier–Stokes Equations: Weak Attractors, Invariant Measures and Large Deviations

“…In order to overcome the difficulties caused by the nonlinear convection term, a class of 3D globally modified Navier-Stokes equations (GMNSE) was introduced in [10], and the existence and uniqueness of weak and strong solutions were proved. After that, the asymptotic behaviors of solutions of 3D GMNSE have also been investigated in [31,36,56] for deterministic cases and in [1,23] for stochastic cases. In addition, we would like to mention another significant modification on the 3D Navier-Stokes equations called tamed 3D Navier-Stokes equations proposed by M. Röckner and X. Zhang in [39], where they investigated the existence and uniqueness of solutions.…”

Invariant measures and statistical solutions of the globally modified magnetohydrodynamics equations

“…for some positive constants C 4 and C 5 , which depend only on O. The proof of (19) follows directly by using the properties of the operator b(•, •, •), the definition of the map F N and the Agmon's inequality. Finally, from the Hölder inequality together with the embedding of H 1 (O) in L 4 (O) and the fact that the H 1 (O)-norm is equivalent to the V 1 -norm, we easily see that there exist C 6 and C 7 , two positive constant depending only on O, such that…”

“…Essentially, it prevent large gradients and rotational dominating the dynamics and leading to explosions. These modified factors violate the basic laws of mechanics, but mathematically speaking, the GMMHDE (1)-( 2) are well-defined system of equations, just like the modified versions of the Navier-Stokes equations (see, for instance, [1,2,9,19]). The existence of solutions for these equations ( 1)-( 2) is obtained in [4].…”