On the Stationary Solutions to 2D g -Navier-Stokes Equations
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On the Stationary Solutions to 2D g-Navier-Stokes Equations
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Dao Trong Quyet1 · Nguyen Viet Tuan2
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Received: 11 August 2015 / Revised: 21 December 2015 / Accepted: 22 March 2016 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016
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Abstract We consider g-Navier-Stokes equations in a two-dimensional smooth bounded domain . First, we study the existence and exponential stability of a stationary solution under some certain conditions. Second, we prove that any unstable steady state can be stabilized by proportional controller with support in an open subset ω ⊂ such that \ω is sufficiently “small.”
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Keywords g-Navier-Stokes equations · Stationary solutions · Stabilizable · Feedback controller
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Mathematics Subject Classification (2010) 35B35 · 35Q35 · 35D35
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1 Introduction
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Let be a bounded domain in R2 with smooth boundary ∂. We consider the following 2D g-Navier-Stokes equations ⎧ ∂u ⎪ ⎪ − νu + (u · ∇)u = ∇p + f in × R+ , ⎪ ⎨ ∂t ∇ · (gu) = 0 in × R+ , (1) ⎪ ⎪ on ∂, ⎪u = 0 ⎩ u(x, 0) = u0 (x), x ∈ ,
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Dao Trong Quyet
[email protected] Nguyen Viet Tuan [email protected] 1
Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam
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Department of Mathematics, Sao Do University, Hai Duong, Vietnam
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D. T. Quyet, N. V. Tuan 18 19 20 21 22 23 24 25 26 27 28
where u = u(x, t) = (u1 , u2 ) is the unknown velocity vector, p = p(x, t) is the unknown pressure, ν > 0 is the kinematic viscosity coefficient, u0 is the initial velocity. The 2D g-Navier-Stokes equations arise in a natural way when we study the standard 3D Navier-Stokes problem in a 3D thin domain g = × (0, g) (see [17]). As mentioned in [16, 17], good properties of the 2D g-Navier-Stokes equations can lead to an initial study of the 3D Navier-Stokes equations in the thin domain g . In the last few years, the existence and long-time behavior of solutions in terms of existence of attractors for 2D g-NavierStokes equations have been studied extensively in both autonomous and non-autonomous cases (see e.g., [1, 2, 7–13, 16, 19] and the references therein). In this paper, we will study the problem of stability and stabilization for strong stationary solutions to (1). To do this, we assume that the function g satisfies the following assumption:
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(G)
g ∈ W 1,∞ () such that 1/2
0 < m0 ≤g(x)≤M0 for all x = (x1 , x2 ) ∈ , and |∇g|∞ < m0 λ1 , 30 31
where λ1 > 0 is the first eigenvalue of the g-Stokes operator in (i.e., the operator A defined in Section 2 below).
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The aim of this paper is twofold. First, we study the existence, uniqueness, and stability of strong stationary solutions to problem (1). The existence of stationary solutions is proved by using the compactness method. When the viscosity is “larger” than the external force, we show that the stationary solution is unique and is globally exponentially stable. In other cases, i.e.
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