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2003 On the solutions of the renormalized equations at all orders
R. M. Temam, D. Wirosoetisno
Adv. Differential Equations 8(8): 1005-1024 (2003).


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 in this paper 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 of the renormalized (approximate) equation. In the dissipative case, we also investigate the higher-order renormalized equations, pursuing [16]: in particular, we show for sufficiently small $\varepsilon$ that the solutions of 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.


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R. M. Temam. D. Wirosoetisno. "On the solutions of the renormalized equations at all orders." Adv. Differential Equations 8 (8) 1005 - 1024, 2003.


Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1038.34057
MathSciNet: MR1989358

Primary: 34C11
Secondary: 34E05, 37C50

Rights: Copyright © 2003 Khayyam Publishing, Inc.


Vol.8 • No. 8 • 2003
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