On the attractor for 2D Navier-Stokes-like system with the dynamic slip boundary condition in a channel

TL;DR

This paper studies the long-term behavior of a 2D fluid flow in a channel with dynamic slip boundary conditions, proving the existence of a global attractor and estimating its complexity based on physical parameters.

Contribution

It introduces a method to construct the global attractor for the 2D Navier-Stokes-like system with dynamic slip boundary conditions and provides an explicit upper bound on its fractal dimension.

Findings

Existence of a unique weak solution that is also strong.

Construction of the global attractor for the system.

Explicit upper bound on the fractal dimension of the attractor.

Abstract

We consider a 2D infinite channel domain with an incompressible fluid satisfying the so-called dynamic slip boundary condition on the (part of the) boundary. Introducing an exhaustion by a sequence of bounded sub-domains of the whole channel we show that the unique weak solution is strong. We then construct the global attractor and find an explicit upper bound of its fractal dimension with regard to the physical parameters. This result is compatible with the analogous estimate in the case of the Dirichlet boundary condition.