Remarks on uniform attractors for the 3D non-autonomous Navier-Stokes-Voight equations
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Remarks on uniform attractors for the 3D nonautonomous Navier-Stokes-Voight equations Yiwen Dou1,2, Xinguang Yang3* and Yuming Qin4 * Correspondence: [email protected] 3 College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China Full list of author information is available at the end of the article
Abstract In this paper, we show the existence of pullback attractors for the non-autonomous Navier-Stokes-Voight equations by using contractive functions, which is more simple than the weak continuous method to establish the uniformly asymptotical compactness in H1 and H2. 2010 Mathematics Subject Classification: 35D05; 35M10 Keywords: Navier-Stokes-Voight equations, processes, contractive functions, uniform attractors
1 Introduction Let Ω ⊂ R3 be a bounded domain with sufficiently smooth boundary ∂Ω. We consider the non-autonomous 3D Navier-Stokes-Voight (NSV) equations that govern the motion of a Klein-Voight linear viscoelastic incompressible fluid: ut − νu − α 2 ut + (u · ∇)u + ∇p = f (t, x), ∇ · u = 0,
x ∈ ,
u(t, x)|∂ = 0, u(τ , x) = uτ (x),
(x, t) ∈ × Rτ ,
t ∈ Rτ ,
t ∈ Rτ , x ∈ ,
(1:1) (1:2) (1:3)
τ ∈ Rτ .
(1:4)
Here u = u(t, x) = (u1(t, x), u2(t, x), u3(t, x)) is the velocity vector field, p is the pressure, ν > 0 is the kinematic viscosity, and the length scale a is a characterizing parameter of the elasticity of the fluid. When a = 0, the above system reduce to the well-known 3D incompressible NavierStokes system: ut − vu + (u · ∇)u + ∇p = f (t, x), x ∈ , t ∈ Rτ ,
(1:5)
∇ · u = 0, x ∈ , t ∈ Rτ .
(1:6)
For the well-posedness of 3D incompressible Navier-Stokes equations, in 1934, Leray [1-3] derived the existence of weak solution by weak convergence method; Hopf [4] improved Leray’s result and obtained the familiar Leray-Hopf weak solution in 1951. Since the 3D Navier-Stokes equations lack appropriate priori estimate and the strong © 2011 Dou et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Dou et al. Boundary Value Problems 2011, 2011:49 http://www.boundaryvalueproblems.com/content/2011/1/49
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nonlinear property, the existence of strong solution remains open. For the infinitedimensional dynamical systems, Sell [5] constructed the semiflow generated by the weak solution which lacks the global regularity and obtained the existence of global attractor of the 3D incompressible Navier-Stokes equations on any bounded smooth domain. Chepyzhov and Vishik [6] investigated the trajectory attractors for 3D nonautonomous incompressible Navier-Stokes system which is based on the works of Leray and Hopf. Using the weak convergence topology of the space H (see below for the definition), Kapustyan and Valero [7] proved the existence of a weak attractor in b
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