The attractors for the nonhomogeneous nonautonomous Navier–Stokes equations.

*(English)*Zbl 1111.35042The 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.

\[ \begin{aligned} & u_t-\nu\Delta u+(u\nabla)u+\nabla p=f\\ & \text{div\,}u=0\text{ on }\Omega,\quad u=\varphi\text{ on }\partial\Omega, \quad \Omega\subseteq \mathbb{R}^2\end{aligned}\tag{1} \] 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.

Reviewer: Bruno Scarpellini (Basel)

##### MSC:

35Q30 | Navier-Stokes equations |

37L30 | Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

##### Keywords:

measure of noncompactness; inhomogeneous Navier-Stokes equations; bounded Lipschitz domain; Stokes operator; existence of a global attractor
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\textit{D. Wu} and \textit{C. Zhong}, J. Math. Anal. Appl. 321, No. 1, 426--444 (2006; Zbl 1111.35042)

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