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On the solutions of the renormalized equations at all orders. (English) Zbl 1038.34057
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.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34C11 Growth and boundedness of solutions to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
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