Quantitative homogenization of attractors of non-Newtonian filtration equations
Summary: For a rapidly spatially oscillating nonlinearity \(g\) we compare solutions \(u^\varepsilon\) of a non-Newtonian filtration equation \[ \partial_t\beta(u^\varepsilon)-D(| Du^\varepsilon|^{p-2}Du^\varepsilon+\varphi(u^\varepsilon)Du^\varepsilon)+ g(x, x/\varepsilon, u^\varepsilon) = f(x, x/\varepsilon) \] with solutions \(u^0\) of the homogenized, spatially averaged equation \[ \partial_t\beta(u^0) - D(| Du^0|^{p-2}Du^0)+\varphi(u^0)Du^0) + g^0(x, u^0)= f^0(x). \] Based on an \(\varepsilon\)-independent a priori estimate, we prove that \[ \|\beta(u^\varepsilon)-\beta(u^0)\|_{L^1(\Omega)} \leq C\varepsilon e^{\rho t} \] uniformly for all \(t\geq 0\). Besides, we give an explicit estimate for the distance between the nonhomogenized \(A^\varepsilon\) and the homogenized \(A^0\) attractors in terms of the parameter \(\varepsilon\); precisely, we show fractional-order semicontinuity of the global attractors for \(\varepsilon\searrow 0: \text{dist}_{L^1(\Omega)}(A^\varepsilon,A^0)\leq C\varepsilon^\gamma\).
scientific article; zbMATH DE number 5728469
scientific article