Preprint No.
A-00-11
Bernold Fiedler, Mark I. Vishik
Quantitative homogenization of analytic semigroups and reaction
diffusion equations with Diophantine spatial frequencies
Abstract:
Based on an analytic semigroup setting, we first consider semilinear reaction
diffusion
equations with spatially quasiperiodic coefficients in the nonlinearity, rapidly
varying
on spatial scale $\varepsilon$. Under periodic boundary conditions, we derive
quantitative
homogenization estimates of order $\varepsilon^\gamma$ on strong Sobolev spaces
$H^\sigma$ in the triangle
$$0 < \gamma < \min (\sigma -n/2,2-\sigma).$$
Here $n$ denotes spatial dimension. The estimates measure the distance to a
solution of
the homogenized equation with the same initial condition, on bounded time
intervals. The
same estimates hold for $C^1$-convergence of local stable and unstable manifolds
of
hyperbolic equilibria.
As a second example, we apply our abstract semigroup result to homogenization of
the
Navier-Stokes equations with spatially rapidly varying quasiperiodic forces in
space
dimensions 2 and 3.
In both examples, a Diophantine condition on the spatial frequencies is crucial
to our
homogenization results. Our Diophantine condition is satisfied for sets of
frequency
vectors of full Lebesgue measure.
In the companion paper \cite{fievis00}, based on $L^2$-methods, these results
are extended
to quantitative homogenization of global attractors in near-gradient
reaction-diffusion
systems.
Keywords:
Mathematics Subject Classification (MSC91):
Language: ENG
Available: Pr-A-00-11.ps
Contact: Bernold Fiedler, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (fiedler@math.fu-berlin.de)
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