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Asymptotic expansion for a periodic boundary condition. (English) Zbl 0849.35043

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 \(\varepsilon\to 0\).

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35C20 Asymptotic expansions of solutions to PDEs
76S05 Flows in porous media; filtration; seepage
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