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Báo cáo toán học: " Remarks on uniform attractors for the 3D nonautonomous Navier-Stokes-Voight equations"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Remarks on uniform attractors for the 3D nonautonomous Navier-Stokes-Voight equations | Dou et al. Boundary Value Problems 2011 2011 49 http www.boundaryvalueproblems.eom content 2011 1 49 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Remarks on uniform attractors for the 3D non-autonomous Navier-Stokes-Voight equations Yiwen Dou1 2 Xinguang Yang3 and Yuming Qin4 Correspondence yangxinguangyxg@yahoo.cn 3College 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 H 1 V and H2. 2010 Mathematics Subject Classification 35D05 35M10 Keywords Navier-Stokes-Voight equations processes contractive functions uniform attractors 1 Introduction Let o c R3 be a bounded domain with sufficiently smooth boundary do. We consider the non-autonomous 3D Navier-Stokes-Voight NSV equations that govern the motion of a Klein-Voight linear viscoelastic incompressible fluid ut vAu u2hu u -V u Vp f t x x t e o X RT 1 1 V- u 0 x e Q t e Rt 1.2 u t x 3Q 0 t e Rt 1.3 u T x uT x x e o T e RT. 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 V 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 Navier-Stokes system ut vAu u V u Vp f t x x e o t e RT 1.5 V u 0 x e o t e RT. 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 .