Asymptotic expansion for a periodic boundary condition
The authors introduce an approximation to flow problems in porous media when the boundary of the flow domain is partially covered by fluid. If the domain is \(\Omega= (0, \pi)\times (0, \pi)\) with a typical point \((x, y)\), they write \(\alpha\) and \(\gamma\) for the subsets of \(\partial\Omega\) on which \(y= 0\) and \(y= \pi\), respectively, and \(\beta= \partial\Omega \backslash (\alpha\cup \gamma)\). Then their approximation can be written as \[ u_t- \Delta u= f(x, y, t)\text{ in } \Omega\times (0, T),\;a\Biggl({x\over \varepsilon}\Biggr) (u_y+ \sigma u+ \psi)+ \Biggl(1- a\Biggl({x\over \varepsilon}\Biggr)\Biggr) u= 0\text{ on } \alpha\times (0, T), \] \[ u_x= 0\text{ on } \beta\times (0, T),\;u= 0\text{ on } \gamma\times (0, T),\;u= \varphi\text{ on } \Omega\times \{0\}, \] where \(a\) is the \(2\pi\)-periodic function which is zero on intervals of length \(\delta< 2\pi\) centered at odd multiples of \(\pi\) and zero elsewhere, \(\psi\) and \(\sigma\) are functions of \((x, t)\), and \(\varepsilon\) is the reciprocal of an even integer. Under suitable regularity hypotheses on the functions \(f\), \(\psi\), \(\sigma\), and \(\varphi\), they study the behavior of the solution \(u_\varepsilon\) as \(\varepsilon\to 0\). Specifically, they show that \(u_\varepsilon= u+ \varepsilon u^1+ \varepsilon R_\varepsilon\), with \(u\) and \(u^1\) solutions of explicitly given initial-boundary value problems and \(R_\varepsilon\) an error term which is bounded in \(L^\infty\) and which tends to zero in any \(L^p\) (\(p\) finite) as \(\varepsilon\to 0\).
scientific article; zbMATH DE number 790886