Preprint No.
A-00-18
Bernold Fiedler, Mark I. Vishik
Quantitative homogenization of global attractors for
reaction-diffusion systems with rapidly oscillating terms
Abstract:
For rapidly spatially oscillating nonlinearities $f$ and
inhomogeneities $g$ we
compare solutions
$u^\varepsilon$ of reaction-diffusion systems
$$\partial_t u^\varepsilon =a\Delta u^\varepsilon
-f(\varepsilon,x,x/\varepsilon,u)+
g(\varepsilon,x,x/\varepsilon)$$
with solutions $u^0$ of the formally
homogenized,
spatially averaged system
$$\partial_t u^0 =a\Delta u^0 -f^0(x,u^0)+g^0(x,u^0).$$
We consider a smooth bounded domain $x\in \Omega\subseteq\mathr^n,\;n\leq
3,$ with Dirichlet boundary conditions. We also impose sufficient regularity and
dissipation conditions, such that solutions exist globally in time and, in fact,
converge
to their compact global attractors ${\cal A}^\varepsilon$ and ${\cal A}^0$,
respectively, in
$L_2(\Omega).$
Based on $\varepsilon$-independent a priori estimates we prove
$$\|u^\varepsilon(\cdot,t)-u^0(\cdot,t)\|_{L_2(\Omega)}\leq C\varepsilon e^{\rho
t},$$
uniformly for all $t\geq 0$ and $0 <\varepsilon\leq \varepsilon_0.$ Here the
solutions
$u^\varepsilon$ and $u^0$ start at the same initial condition $u=u_0(x)\in
H^1(\Omega)$ for
$t=0,$ and
$C=C(\|u_0\|_{H^1}).$
Based on an $\varepsilon$-independent $H^2$-bound on the global attractors
${\cal
A}^\varepsilon$ as well as an exponential attraction rate $\nu$ of the
homogenized attractor
${\cal A}^0$ in
$L_2(\Omega),$ we also prove fractional order upper semicontinuity of the global
attractors for $\varepsilon\searrow 0,$
$$dist_{L_2(\Omega)}({\cal A}^\varepsilon,{\cal A}^0)\leq
C\varepsilon^{\gamma'}$$
for $\gamma'=(1+\rho/\nu)^{-1}.$
This result requires the homogenized nonlinearity $f^0(x,w)$ to be near a
potential vector
function $f^1(x,w)=\nabla_w F(x,w)$ with scalar potential $F.$
Both quantitative homogenization estimates are proved only for quasiperiodic
dependence of
$f,g$ on the spatially rapidly oscillating variable $x/\varepsilon.$ Moreover,
the finitely many frequencies describing this quasiperiodic dependence are
assumed to
satisfy Diophantine conditions, as are familiar from small divisor problems in
Kolmogorov-Arnold-Moser theory. Alternatively, the results hold if $f,g$ admit a
sufficiently regular divergence representation which describes their explicit
spatial
dependence. All results apply to, and are illustrated for, the case of
FitzHugh-Nagumo
systems with spatially rapidly oscillating quasiperiodic coefficients and
inhomogeneities.
In the companion paper \cite{fievis00a}, based on analytic semigroup methods,
similar
results are obtained for the quantitative homogenization of solutions and
invariant
manifolds. Examples include homogenization of the Navier-Stokes system under
periodic
boundary conditions and for spatially rapidly oscillating quasiperiodic forces.
Keywords:
Mathematics Subject Classification (MSC91):
Language: ENG
Available: Pr-A-00-18.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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