Ginzburg-Landau equation and motion by mean curvature. II: Development of the initial interface
[For part I, see ibid., 437-475 (1997; Zbl 0935.35062).] The paper deals with the study of the short time behavior of solutions to the Ginzburg-Landau equation \(u_t^\varepsilon - \Delta u^\varepsilon + 2/\varepsilon^2 u^\varepsilon (u^\varepsilon -1) (u^\varepsilon +1)=0\) in \({\mathbb R}^d\times [0,+\infty)\), with the initial condition \(u^\varepsilon (x,0)=u_0^\varepsilon (x)\). The main result of the work establishes that under appropriate assumptions on \(u_0^\varepsilon\), solutions converge to \(\pm 1\) in short time and, they behave like the one-dimensional traveling wave along the interface. As a consequence of this result, the author deduces that the energy remains uniformly bounded with respect to \(\varepsilon\). The main results are proved after deducing two key estimates: (i) a detailed description of \(u^\varepsilon\) near the initial interface, which is a sharper version of a previous result of DeMottoni and Schatzman, and (ii) a gradient estimate (Theorem 5.1 in the paper).
scientific article; zbMATH DE number 1326049