Dynamic Behavior of the Three-Dimensional Globally Modified Navier–Stokes Equations With Memory Term

Abstract

This paper focuses on investigating the dynamic behavior of the three-dimensional globally modified Navier–Stokes equations with memory term, based on the general framework of dynamical systems arising from differential equations with delay terms. The method we employed diverges from previous approaches investigating the dynamic behavior of equations with memory term by Dafermos transformations. First, we use the classical Galerkin method to prove the well-posedness of the solutions to the transformation equation. Subsequently, by constructing an appropriate linear mapping, we demonstrated the existence of solutions to the original equation, thereby the semigroup generated by the original equation is established in phase space $V\times L_{D(A)}^2$ . Lastly, by showing the existence of bounded absorbing sets for the semigroup and its asymptotic compactness, the existence of global attractors is proved.