A rapidly converging phase field model
The authors introduce asymptotic expansion of the solution of a phase field problem in order to simplify its resolution. The authors consider the coupled evolution problem \(\varepsilon ^{2}(\alpha _{\varepsilon }\varphi _{\varepsilon t}-\Delta \varphi _{\varepsilon })+W^{\prime }(\varphi _{\varepsilon })=\varepsilon mG^{\prime }(\varphi _{\varepsilon })u_{\varepsilon }\) ; \((u_{\varepsilon }+l\varphi _{\varepsilon }/2)_{t}=\Delta u_{\varepsilon }\), where \(\varphi _{\varepsilon }\) (resp. \(u_{\varepsilon }\)) is the phase indicator (between \(-1\) and \(+1\)) (resp. the temperature). \(W\) is a smooth double-well potential and \(G\) is a smooth function, both satisfying usual conditions. The authors assume that \(\alpha _{\varepsilon }=\alpha +\varepsilon \alpha _{1}\), where \( \alpha \) and \(\alpha_{1}\) are positive constants, \(\alpha _{1}\) being supposed to be linked to \(G\) and to the data of the problem. The main purpose of the paper is to introduce inner and outer asymptotic expansions of the solution, according to the proximity of \(\varphi _{\varepsilon }\) to \( \pm 1\), to compute the two first order terms of this asymptotic expansions and to prove, under further assumptions, that the difference between the solution and these two first-order terms is of order \(O(\varepsilon ^{2})\). The paper ends with a 1D-example.
scientific article; zbMATH DE number 5141118