Feedback Control of Navier-Stokes-Voigt Equations by Finite Determining Parameters
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Feedback Control of Navier-Stokes-Voigt Equations by Finite Determining Parameters Nguyen Thi Ngan1 · Vu Manh Toi2 Received: 22 January 2019 / Revised: 15 April 2019 / Accepted: 25 April 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We study the stabilization of stationary solutions to Navier-Stokes-Voigt equations by finitedimensional feedback control scheme introduced by Azouani and Titi (Evol. Equ. Control Theory 3, 579–594 2014). The designed feedback control scheme is based on the finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and volume elements. Keywords Navier-Stokes-Voigt equations · Feedback control · Stabilization · Determining modes · Determining nodes · Determining finite volume Mathematics Subject Classification (2010) 76A10 · 93D15 · 35Q35 · 93C20
1 Introduction Let = (0, L)d , L > 0, be a periodic box in Rd , d ∈ {2, 3}. We consider the following Navier-Stokes-Voigt equations: ∂t (y − α 2 y) − νy + (y · ∇)y + ∇p = f (x), (1) ∇ ·y =0 for all (x, t) ∈ × (0, ∞), subject to the periodic boundary conditions: y(x, t) = y(x + L, t), x ∈ , t > 0, Vu Manh Toi
[email protected] Nguyen Thi Ngan [email protected] 1
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2
Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam
(2)
N.T. Ngan, V.M. Toi
and the initial condition y(x, 0) = y 0 ,
x ∈ ,
(3)
where y = y(x, t) is the unknown velocity and p = p(x, t) is the unknown pressure, ν > 0 is the kinematic viscosity coefficient, and α is a length-scale parameter. The system (1) was introduced by Oskolkov in [22] as a model of motion of certain linear viscoelastic fluids. The system (1) was also proposed by Cao, Lunasin, and Titi in [8] as a regularization, for small value of α, of the 3D Navier-Stokes equations for the sake of direct numerical simulations. In fact, this system belongs to the so-called α-models in fluid mechanics (see, e.g., [14] for other models in this class). We also refer the reader to [12] for an interesting application of Navier-Stokes-Voigt equation in image inpainting. In the last few years, mathematical questions related to Navier-Stokes-Voigt equations have attracted the attention of a number of mathematicians. The existence and long-time behavior of solutions in terms of existence of attractors to the Navier-Stokes-Voigt equations in domains that are bounded or unbounded but satisfying the Poincar´e inequality was investigated extensively in [3, 9, 11, 13, 15, 16, 24]. The time decay rates of solutions to the equations on the whole space were studied in [4, 21, 25]. Recently, the stabilization of stationary solutions to the 3D Navier-Stokes-Voigt by using internal feedback controls was studied in [5], and the effect of fast oscillating-in-time forces on the long-time behavior of solutions
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