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

Abstract

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 τ ". 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.

Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

“…Viscoelastic materials are those which exhibit both viscous and elastic characteristics, when undergoing deformation and non-Newtonian fluids are those which does not follow Newton's law of viscosity. For the past several decades, the mathematical theory of non-Newtonian and viscoelastic fluid flows have been developed by several mathematicians starting from the works of Oskolkov (see for example [2,11,18,26,30,31,32,33,34,41], etc and the references therein). In this work, we consider a linear viscoelastic fluid with a relaxation time λ and retardation times {κ 1 , κ 2 }.…”

“…The global solvability results of the system (1) in bounded domains are available in the literature, cf. [30,33,41], etc. The 2D Navier-Stokes-Voight equation in an unbounded strip-like domain is considered in [3] and the authors established that the semigroup generated by this equation has a global attractor in weighted Sobolev spaces.…”

Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

“…From estimates (19) and (20) and Lemma 2, it follows that there exist a subsequence {m k } ∞ k=1 and a function u such that v m k converges strongly to u in the space C([0, T]; V 1 (Ω)) as k → ∞. Without loss of generality, we can assume that…”

“…Artemov and Baranovskii [18] proved the existence of weak solutions to the coupled system of nonlinear equations describing the heat transfer in steady-state flows of a polymeric fluid. Mohan [19] investigated the global solvability, the asymptotic behavior, and some control problems for the NSV model with "fading memory" and "memory of length τ".…”

Approximate controllability of a non-autonomous evolution equation in Banach spaces

“…From the mathematical point of view, in infinite dimensions, the problems of exact and approximate controllability are to be distinguished. Exact controllability enables to steer the system to an arbitrary final state (see [35] for the exact controllability of a Galerkin approximated system), while approximate controllability means that the system can be steered to an arbitrary small neighborhood of a final state. The literature available on the infinite dimensional control systems also pointed out that the exact controllability rarely holds (cf.…”