On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations

Manil T. Mohan

doi:10.3934/eect.2020007

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2020, Volume 9, Issue 2: 301-339. Doi: 10.3934/eect.2020007

This issue Previous Article Next Article Regularized solution for a biharmonic equation with discrete data

On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations

Manil T. Mohan^,

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

^*Corresponding author: Manil T. Mohan
^*Corresponding author: Manil T. Mohan

Received: November 30, 2018

Revised: January 31, 2019

Early access: August 2019

Published: June 2020

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In this work, we consider the three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt fluids in bounded and unbounded domains. We investigate the global solvability results, asymptotic behavior and also address some control problems of such viscoelastic fluid flow equations with "fading memory" and "memory of length $ \tau $". A local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique are used to obtain global solvability results. Since we are not using compactness arguments in the proofs, the global solvability results are also valid in unbounded domains like Poincaré domains. We also remark that using an $ m $-accretive quantization of the linear and nonlinear operators, one can establish the existence and uniqueness of strong solutions for the Navier-Stokes-Voigt equations and avoid the tedious Galerkin approximation scheme. We examine the asymptotic behavior of the stationary solutions and also establish the exponential stability results. Finally, under suitable assumptions on the Galerkin basis, we consider the controlled Galerkin approximated 3D Kelvin-Voigt fluid flow equations. Using the Hilbert uniqueness method combined with Schauder's fixed point theorem, the exact controllability of the finite dimensional Galerkin approximated system is established.

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Mathematics Subject Classification: Primary: 49J20; Secondary: 35Q35, 76D03.

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