Local null controllability of a class of non-Newtonian incompressible viscous fluids

TL;DR

This paper establishes the null controllability of certain non-Newtonian incompressible viscous fluid models, extending techniques to shear-dependent viscosities and providing a computational algorithm with numerical validation.

Contribution

It develops a controllability framework for non-Newtonian fluids with shear-dependent viscosity and introduces a convergent quasi-Newton algorithm for control computation.

Findings

Existence of smooth controls for linearized non-Newtonian fluid systems.

Extension of controllability results to nonlinear models.

Numerical experiments demonstrating the control algorithm's effectiveness.

Abstract

We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.