Global existence for a 3D non-stationary Stokes flow with Coulomb's type friction boundary conditions

TL;DR

This paper establishes the existence and uniqueness of solutions for 3D non-stationary Stokes flows with Coulomb's friction boundary conditions, providing sharp estimates and handling non-local friction laws.

Contribution

It introduces a rigorous mathematical framework for non-stationary Stokes flows with nonlinear Coulomb friction boundary conditions, including non-local cases, with existence, uniqueness, and estimate results.

Findings

Proved existence and uniqueness for shear stress below threshold.

Derived sharp estimates for velocity, pressure, and stress tensor.

Extended results to non-local Coulomb friction case using iterative approximation.

Abstract

In this paper we study non stationary viscous incompressible fluid flows with nonlinear boundary slip conditions given by a subdifferential property of friction type. More precisely we assume that the tangential velocity vanishes as long as the shear stress remains below a threshold, that may depend on the time and the position variables but also on the stress tensor, allowing to consider Coulomb's type friction laws. An existence and uniqueness theorem is obtained first when the shear stress threshold is a data and sharp estimates are derived for the velocity and pressure fields as well as for the stress tensor. Then an existence result is proved for the non-local Coulomb's friction case by using a successive approximation technique with respect to the shear stress threshold.