Homogenization of the Stokes equations with high-contrast viscosity.
This paper deals with the homogenization of the Stokes equations in a vertical open cylinder \(\Omega=\Omega^\prime\times (0,1)\) of \(\mathbb R^3\) with varying viscosity and with Dirichlet boundary condition: \[ \begin{cases} -\text{div}(a_\varepsilon \nabla u^\varepsilon)-\nabla p_\varepsilon=f &\text{ in } \Omega,\\ \text{div} (u^\varepsilon)=0 &\text{ in } \Omega,\\ u^\varepsilon=0&\text{ on } \partial\Omega. \end{cases} \] The viscosity \(a_\varepsilon\) is \(\varepsilon\)-periodic, \(a_\varepsilon\) takes a constant value \(\alpha_\varepsilon\gg 1\) in a \(\varepsilon\)-periodic lattice \(\omega_\varepsilon\) of vertical cylinders of radius \(\varepsilon r_\varepsilon\), where \( r_\varepsilon<<1\), and is equal to \(1\) outside \(\omega_\varepsilon\). In the critical regime defined by \(\lim_{\varepsilon\to 0}\varepsilon^2| \ln r_\varepsilon| \in (0,\infty)\) and \(\lim_{\varepsilon\to 0}\alpha_\varepsilon r_\varepsilon^2 \in (0,\infty]\), the limit problem is a coupled Stokes system satisfied by the limit velocity and the limit of the rescaled velocity in the cylinders, which can be read as a nonlocal law of Brinkman type. Moreover, if \(\lim_{\varepsilon\to 0}\alpha_\varepsilon r_\varepsilon^2 =+\infty\), the limit of the rescaled velocity is equal to zero and the Brinkman law is derived as in a paper by \textit{G. Allaire} [in: Appl. Math. Engin. Sci., Proc. 2nd Fr.-Chil. Symp., Santiago de Chile 1989, 7--20 (1991; Zbl 0759.76072)]. In the other regimes the homogenization leads either to classical Stokes problems or to a zero limit velocity. In the critical case the pressure is not bounded in \(L^2\) but only in \(H^{-1}\). Moreover the pressure of the limit problem is not equal to the weak limit of the pressure in \(H^{-1}\).
scientific article; zbMATH DE number 2006034
scientific article