On the existence of an exponential attractor for a planar shear flow with Tresca's friction condition

TL;DR

This paper proves the existence of a finite-dimensional global attractor and an exponential attractor for a 2D Navier-Stokes shear flow with Tresca boundary conditions, using the method of l-trajectories.

Contribution

It establishes the existence of exponential attractors for a nonstationary shear flow with Tresca's friction law, extending attractor theory to this boundary condition.

Findings

Existence of a global attractor of finite fractional dimension.

Existence of an exponential attractor for the flow.

Application of l-trajectories method to boundary conditions with Tresca law.

Abstract

We consider a two-dimensional nonstationary Navier-Stokes shear flow with a subdifferential boundary condition on a part of the boundary of the flow domain, namely, with a boundary driving subject to the Tresca law. There exists a unique global in time solution of the considered problem which is governed by a variational inequality. Our aim is to prove the existence of a global attractor of a finite fractional dimension and of an exponential attractor for the associated semigroup. We use the method of $l$-trajectories. This research is motivated by a problem from lubrication theory.