## Advances in Differential Equations

### On the solutions of the renormalized equations at all orders

#### Abstract

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.

#### Article information

Source
Adv. Differential Equations, Volume 8, Number 8 (2003), 1005-1024.

Dates
First available in Project Euclid: 19 December 2012

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