Attractors for weak and strong solutions of the three-dimensional Navier-Stokes equations with damping

TL;DR

This paper establishes the existence and regularity of global attractors for weak and strong solutions of the 3D Navier-Stokes equations with damping, depending on parameter values, and includes numerical simulations.

Contribution

It proves the existence of global attractors for weak solutions with damping and shows their regularity and relation to strong solutions under specific parameter conditions.

Findings

Existence of global attractors for weak solutions with damping.

Regularity of attractors for certain parameter ranges.

Numerical simulations illustrating theoretical results.

Abstract

In this paper we obtain the existence of global attractors for the dynamical systems generated by weak solution of the three-dimensional Navier-Stokes equations with damping. We consider two cases, depending on the values of the parameters \b{eta},{\alpha} controlling the damping term and the viscosity {\mu}. First, for \b{eta} we define a multivalued dynamical systems and prove the existence of the global attractor as well. Second, for either \b{eta}>3 or \b{eta}=3, 4{\alpha}{\mu}>1 the weak solutions are unique and we prove that the global attractor for the corresponding semigroup is more regular. Also, we prove in this case that it is the global attractor for the semigroup generated by the strong solutions. Finally, some numerical simulations are performed.