an:02002556
Zbl 1038.34057
Temam, R. M.; Wirosoetisno, D.
On the solutions of the renormalized equations at all orders.
EN
Adv. Differ. Equ. 8, No. 8, 1005-1024 (2003).
1079-9389
2003
j
34G20 34C11 34E05 34C29 34E13
Summary: The renormalization (or averaging) procedure is often used to construct approximate solutions in evolutionary problems with multiple timescales arising from a small parameter \(\varepsilon\). We show here that the leading-order approximation shares two important properties of the original system, namely energy conservation in the inviscid case and dissipation rate (coercivity) in the forced-dissipative case. This implies the boundedness of the solutions to the renormalized (approximate) equation. In the dissipative case, we also investigate the higher-order renormalized equations, pursuing [the authors, Appl. Math. Optimization 46, 313--330 (2002; Zbl 1031.34052)]; in particular, we show, for sufficiently small \(\varepsilon\), that the solutions to these equations are bounded and that the dissipativity property of the original system carries over in a modified form. This is shown by a simple estimate based on the above leading-order result, and, alternatively, by a ``shadowing'' argument.
1031.34052