The validity of phase diffusion equations and of Cahn-Hilliard equations for the modulation of pattern in reaction-diffusion systems (Q2642064)

Temporarily and spatially slow modulations of stable (as well as slightly unstable) spatially periodic patterns are typically described within the framework of phase diffusion and Cahn-Hilliard equations, where these evolutionary PDEs are derived via a multiple scaling analysis. This paper deals with the validity of such equations to capture the above pattern formation processes. Thereto, estimates between formal approximations and true solutions are deduced, with proofs given for an abstract reaction-diffusion equation as original system. In particular, one finds validity results for the phase diffusion and the Cahn-Hilliard equation. The interesting approach is based on techniques used in [\textit{A. Doelmann, B. Sandstede, A. Scheel} and \textit{G. Schneider}, The dynamics of modulated wave trains. Mem. Am. Math. Soc. 934, 105 p. (2009; Zbl 1179.35005)], like appropriate coordinate transformations, a Bloch-wave analysis and a separation into critical and noncritical modes.