# zbMATH — the first resource for mathematics

The attractors for the nonhomogeneous nonautonomous Navier–Stokes equations. (English) Zbl 1111.35042
The authors consider the inhomogeneous Navier-Stokes equations
\begin{aligned} & u_t-\nu\Delta u+(u\nabla)u+\nabla p=f\\ & \text{div\,}u=0\text{ on }\Omega,\quad u=\varphi\text{ on }\partial\Omega, \quad \Omega\subseteq \mathbb{R}^2\end{aligned}\tag{1}
on a bounded Lipschitz domain $$\Omega$$ in $$\mathbb R^2$$. One assumes
$f=f(x,t)\in{\mathcal L}^2_{\text{loc}}((0,T),E),\quad \varphi\in{\mathcal L}^2(\partial\Omega)\tag{2}$ where $$E=\text{dom}(A^{\frac14})$$, with $$A=-P_\Delta$$ the Stokes operator associated with (1). The aim is to prove the existence of a global attractor for (1). To do so, the authors need several preparatory steps. First, using a suitable background flow $$\psi$$, eq. (1) is transformed into a new one, based on Dirichlet boundary conditions, i.e.:
$v_t+\nu Av+B(v,v)+B(v,\psi)+B(\psi,v)=P(f+\nu F)-B(\psi,\psi)\tag{3}$
where $$F$$ is an additional force term induced by the background flow $$\psi$$. In order to prove the existence of an attractor for (3), the authors have to rely on work of V. V. Chepyzhov and M. I. Vishik [Am. Math. Soc. Colloq. Publ. 49, 363 p. (2002; Zbl 0986.35001)]; they introduce a number of notions and discuss their properties. Thus one has the notion of indexed process $$\{U_\sigma(t,\tau)\mid t\geq \tau,\;\tau\in\mathbb R, \sigma\in\Sigma\}$$ where $$\Sigma$$ is the index space (a metric space), $$\sigma$$ the symbol of the process and $$\{U_\sigma(t,\tau)\}$$ a family of mappings on a Banach space $$E$$ such that
$U(t,s)U(s,\tau)=U(t,\tau),\quad U(\tau,\tau)=\text{Id},\quad t\geq s\geq \tau,\quad \tau\in\mathbb R.$
In terms of this notion, the relevant topological concepts such as absorbing set, $$\omega$$-limit set, uniform attractor etc. are introduced, and some of their properties summarized. Criteria (Thms. 4.1, 4.2) for the existence of a uniform attractor are given. Finally, the index space is made precise: it is based on the translates $$(T_hf)(s)=f(h+s)$$ induced by the exterior force $$f$$ in (1) resp. (3). In the main section 6 the existence of a uniform attractor in the sense of Chepyzhov and Vishik (loc. cit.) is proved. First, it is noted that existence of global solutions of (3) is guaranteed by a Galerkin method; for details the reader is referred to R. M. Brown, P. A. Perry, and Zh. Shen [Indiana Univ. Math. J. 49, 81–112 (2000; Zbl 0969.35105)] where a proof in a comparable situation is given. Then one proceeds to the proof of the main Theorem 6.1 which asserts the existence of a uniform attractor for (3). The proof involves lengthy estimates, based in part on the paper of Brown, Perry, Shen (loc. cit.). Theorem 6.2 finally asserts that if $$f(x,s)$$ is translation compact in $$D(A^{-\frac14})$$ then the attractor in question is compact.

##### MSC:
 35Q30 Navier-Stokes equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text:
##### References:
 [1] Ball, J., Global attractors for damped semilinear wave equations, Discrete contin. dyn. syst., 10, 31-52, (2004) · Zbl 1056.37084 [2] Bona, J.L.; Dougalis, V.A., An initial and boundary value problem for a model equation for propagation of long waves, J. math. anal. appl., 75, 503-522, (1980) · Zbl 0444.35069 [3] Brown, R.; Perry, P.; Shen, Z., On the dimension of attractor for the non-homogeneous navier – stomes equations in non-smooth domains, Indiana univ. math. J., 49, 81-112, (2000) · Zbl 0969.35105 [4] Chepyzhov, V.V.; Vishik, M.I., Non-autonomous evolutionary equations with translation compact symbols and their attractors, C. R. acad. sci. Paris Sér. I, 321, 153-158, (1995) · Zbl 0837.35059 [5] Chepyzhov, V.V.; Vishik, M.I., Trajectory attractors for reaction-diffusion systems, Topol. methods nonlinear anal., 7, 49-76, (1996) · Zbl 0894.35010 [6] Chepyzhov, V.V.; Vishik, M.I., Evolution equations and their trajectory attractors, J. math. pures appl., 76, 913-964, (1997) · Zbl 0896.35032 [7] Chepyzhov, V.V.; Vishik, M.I., Attractors for equations of mathematical physics, Amer. math. soc. colloq. publ., vol. 49, (2002), Amer. Math. Soc. Providence, RI · Zbl 0986.35001 [8] Chepyzhov, V.V.; Vishik, M.I., On non-autonomous sine – gordon type equations with a simple global attractor and some averaging, Discrete contin. dyn. syst., 12, 27-38, (2005) · Zbl 1067.35017 [9] 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 [10] Hale, J.K., Asymptotic behavior of dissipative systems, Math. surveys monogr., vol. 25, (1988), Amer. Math. Soc. Providence, RI · Zbl 0642.58013 [11] Haraux, A., Systèmes dynamiques dissipatifs et applications, (1991), Masson Paris · Zbl 0726.58001 [12] Karch, G., Asymptotic behavior of solutions to some pseudoparabolic equations, Math. methods appl. sci., 21, 271-289, (1997) · Zbl 0869.35057 [13] Karachlios, N.I.; Stavrakakis, N.M., Existence of global attractors for semilinear dissipative wave equations on $$R^N$$, J. differential equations, 157, 183-205, (1999) [14] Ladyzhenskaya, O.A., Attractors for semigroups and evolution equations, (1991), Leizioni Lincei, Cambridge Univ. Press Cambridge, New York · Zbl 0729.35066 [15] S.S. Lu, H.Q. Wu, C.K. Zhong, Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces, submitted for publication · Zbl 1083.35094 [16] 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 [17] Miranville, A.; Wang, X., Upper bound on the dimension of the attractor for nonhomogeneous navier – stokes equations, Discrete contin. dyn. syst., 2, 95-110, (1996) · Zbl 0949.35112 [18] Moise, I.; Rosa, R.; Wang, X., Attractors for noncompact semigroups via energy equations, Nonlinearity, 11, 1369-1393, (1998) · Zbl 0914.35023 [19] Moise, I.; Rosa, R.; Wang, X., Attractors for noncompact non-autonomous systems via energy equations, Discrete contin. dyn. syst., 10, 473-496, (2004) · Zbl 1060.35023 [20] Robinson, J.C., Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors, (2001), Cambridge Univ. Press · Zbl 0980.35001 [21] Sell, G.R.; You, Y., Dynamics of evolutionary equations, (2002), Springer New York · Zbl 1254.37002 [22] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1997), Springer New York · Zbl 0871.35001 [23] Zelik, S., The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete contin. dyn. syst., 7, 593-641, (2001) · Zbl 1153.35311
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.