Attractors for Navier-Stokes flows with multivalued and nonmonotone subdifferential boundary conditions

TL;DR

This paper establishes the existence of solutions and attractors for 2D Navier-Stokes flows with complex boundary conditions involving multivalued and nonmonotone subdifferentials, relevant to fluid control in semipermeable domains.

Contribution

It introduces a novel analysis of Navier-Stokes equations with multivalued boundary conditions, proving existence of solutions and attractors for such complex systems.

Findings

Existence of global in time solutions for the PDE inclusion.

Existence of trajectory and weak global attractors.

Application to control problems with semipermeable walls.

Abstract

We consider two-dimensional nonstationary Navier-Stokes shear flow with multivalued and nonmonotone boundary conditions on a part of the boundary of the flow domain. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion with a multivalued term in the form of Clarke subdifferential. Then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow. This research is motivated by control problems for fluid flows in domains with semipermeable walls and membranes.