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Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces. (English) Zbl 1112.35029
The purpose of this paper is to the investigate uniform attractor for the nonautonomous two-dimensional Navier-Stokes equations of viscous incompressible fluid on a bounded domain: $$\frac{\partial u}{\partial t}-\nu\Delta u+(u\cdot \nabla)u+\nabla p=f(x,t),\text{ div}u=0\text{ in }\Omega,\ u=0\text{ on }\partial\Omega, \ u|_{t=\tau}=u_\tau,\ \tau\in\bbfR,\tag 1$$ where $\nu>0$ is the kinematic viscosity of the flow and $f(x,t)$ is the external force. As usually $H$ and $V$ denote the Hilbert spaces \align H&=\bigl\{u\in L^2 (\Omega)^2:\operatorname{div}\,u=0,\ u\cdot\vec n|_{\partial\Omega}=0 \bigr\},\\ V&= \bigl\{u\in H^1_0(\Omega)^2:\operatorname{div}\,u=0\bigr\} \endalign which are endowed with the scalar products and norms of $L^2 (\Omega)^2$ and $H^1_2(\Omega)^2$ respectively. First, the existence and structure of uniform attractors in $H$ is proved for equations with normal external forces in $L^2_{\text{loc}}(\bbfR;V')$. Then, the properties of kernel sections are studied. Finally, the fractal dimension of the kernel sections of the uniform attractor is estimated.

MSC:
 35B41 Attractors (PDE) 35Q30 Stokes and Navier-Stokes equations 76D05 Navier-Stokes equations (fluid dynamics)
Full Text:
References:
 [1] Babin, A.; Vishik, M.: Attractors of evolutionary partial differential equations and estimates of their dimensions. Uspekhi mat. Nauk 38, 133-187 (1983) [2] Babin, A.; Vishik, M.: Attractors for evolution equations. (1992) · Zbl 0778.58002 [3] Chepyzhov, V.; Ilyin, A.: On the fractal dimension of invariant sets: application to Navier -- Stokes equations. Discrete contin. Dyn. syst. 10, 117-135 (2004) · Zbl 1049.37047 [4] Chepyzhov, V.; Vishik, M.: Attractors of nonautonomous dynamical systems and their dimension. J. math. Pures appl. 73, 279-333 (1994) · Zbl 0838.58021 [5] Chepyzhov, V.; Vishik, M.: Trajectory attractors for evolution equations. C. R. Acad. sci. Paris sér. I math. 321, 1309-1314 (1995) · Zbl 0843.35038 [6] Chepyzhov, V.; Vishik, M.: Trajectory attractors for 2D Navier -- Stokes systems and some generalizations. Topol. methods nonlinear anal. 8, 217-243 (1996) · Zbl 0894.35011 [7] Chepyzhov, V.; Vishik, M.: Evolution equations and their trajectory attractors. J. math. Pures appl. 76, 913-964 (1997) · Zbl 0896.35032 [8] Chepyzhov, V.; Vishik, M.: Attractors for equations of mathematical physics. Amer. math. Soc. colloq. Publ. 49 (2002) · Zbl 0986.35001 [9] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040 [10] Efendiev, M.; Zelik, S.: The attractor for a nonlinear reaction -- diffusion system in an unbounded domain. Comm. pure appl. Math. 54, 625-688 (2001) · Zbl 1041.35016 [11] Foias, C.; Temam, R.: Structure of the set of stationary solutions of the Navier -- Stokes equations. Comm. pure appl. Math. 30, 149-164 (1977) · Zbl 0335.35077 [12] Foias, C.; Temam, R.: Some analytic and geometric properties of the solutions of the Navier -- Stokes equations. J. math. Pures appl. 58, No. 3, 339-368 (1979) · Zbl 0454.35073 [13] Haraux, A.: Attractors of asymptotically compact process and applications to nonlinear partial differential equations. Comm. partial differential equations 13, No. 11, 1383-1414 (1988) · Zbl 0676.35008 [14] Haraux, A.: Systèmes dynamiques dissipatifs et applications. (1991) · Zbl 0726.58001 [15] Hale, J.: Asymptotic behavior of dissipative systems. Math. surveys monogr. 25 (1988) · Zbl 0642.58013 [16] Hundertmark, D.; Laptev, A.; Weidl, T.: New bounds on the Lieb -- Thirring constants. Invent. math. 140, 693-704 (2000) · Zbl 1074.35569 [17] Ilyin, A.: Lieb -- Thirring inequalities on the N-sphere and in the plane and some applications. Proc. London math. Soc. (3) 67, 159-182 (1993) · Zbl 0789.58079 [18] Ilyin, A.: Attractors for Navier -- Stokes equations in domain with finite measure. Nonlinear anal. 27, No. 5, 605-616 (1996) · Zbl 0859.35090 [19] Ladyzhenskaya, O.: On the dynamical system generated by the Navier -- Stokes equations. Zap. nauchn. Sem. LOMI 27, 91-115 (1972) · Zbl 0327.35064 [20] Laptev, A.; Weidl, T.: Sharp Lieb -- Thirring inequalities in high dimensions. Acta math. 184, 87-111 (2000) · Zbl 1142.35531 [21] Lieb, E.; Thirring, W.: Inequalities for the moments of the eigenvalues of Schrödinger equations and their relations to Sobolev inequalities. Essays in honour of valentine Bargmann, stud. Math. phys., 269-303 (1976) [22] S. Lu, Attractors for nonautonomous reaction -- diffusion systems with symbols without strong translation compactness, submitted for publication · Zbl 1139.35028 [23] Lu, S.; Wu, H.; Zhong, C.: Attractors for nonautonomous 2D Navier -- Stokes equations with normal external forces. Discrete contin. Dyn. syst. 13, 701-719 (2005) · Zbl 1083.35094 [24] Ma, Q.; Wang, S.; Zhong, C.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana univ. Math. J. 51, 1541-1559 (2002) · Zbl 1028.37047 [25] Robinson, J.: Infinite-dimensional dynamical systems. An introduction to dissipative parabolic pdes and the theory of global attractors. (2001) · Zbl 0980.35001 [26] Rosa, R.: The global attractor for the 2D Navier -- Stokes flow on some unbounded domains. Nonlinear anal. 32, 71-85 (1998) · Zbl 0901.35070 [27] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. (1997) · Zbl 0871.35001 [28] Temam, R.: Navier -- Stokes equations, theory and numerical analysis. (2001) · Zbl 0981.35001 [29] Zelik, S.: The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete contin. Dyn. syst. 7, 593-641 (2001) · Zbl 1153.35311 [30] Zhong, C.; Sun, C.; Niu, M.: On the existence of the global attractor for a class of infinite-dimensional nonlinear dissipative dynamical systems. Chinese ann. Math. ser. B 26, No. 3, 1-8 (2005) · Zbl 1079.35026