Crystalline flow starting from a general polygon (Q2078370)

In this paper, a singular initial value problem for the following system \[ V-j(t)= f_j(\kappa_j/ L_j(t)) \] is investigated, which describes a polygonal flow called crystalline flow. The authors prove existence and uniqueness of the crystalline flow in the case when \(f_j\) is locally Lipschitz and non-decreasing, and \(\lim_{|x|\to\infty}\frac{f_j(x)}{x}=\lambda_j>0\), where \(\lambda_j\) depend on \(j\) and on a finite subset \(\mathcal{N}\) of the unit circle. Moreover, in the paper it is proved that the unique local-in-time crystalline flow converges to \(\Gamma_0\), as \(t\) goes to \(0\), in the Hausdorff distance. Here \(\Gamma_0\) is a polygon whose orientations belong to \(\mathcal{N}\). The authors use a self-similar expanding solution construction to prove the main results in the paper.