Curve shortening flow in heterogeneous media.

*(English)*Zbl 1261.35079This paper concerns the curvature shortening flow of planar curves in a heterogeneous medium, which is modeled by a spatially-dependent additive forcing term:
\[
v = (\kappa +g)\nu,
\]
where \(\nu\) is the inward normal vector to the curve, \(\kappa\) is the curvature, \(v\) is the normal velocity vector, and \(g \in L^\infty(\mathbb R^2)\) represents the forcing term. This comes from a homogenization problem related to the averaged behavior of an interface moving by curvature plus a rapidly oscillating forcing term \(g\left(\frac x\varepsilon,\frac y\varepsilon\right)\), where \(g\) is a \(1\)-periodic Lipschitz continuous function. When \(g\) is periodic, this equation was studied by N. Dirr et al. [Eur. J. Appl. Math. 19, No. 6, 661–699 (2008; Zbl 1185.53076)]. They proved existence and uniqueness of planar pulsating waves in every direction of propagation.

In the present paper, the authors classify all possible singularities which can arise during the evolution and show that, when \(g\) is smooth and the initial curve is embedded, the existence time of a regular solution is bounded below by a quantity depending only on \(\| g\|_\infty\) and on the initial curve. Since the authors have no estimates on the curvature in terms of \(\| g\|_\infty\), they are not able to obtain a general existence result for \(g \in L^\infty\). However, by assuming that the initial curve is the graph of a function \(u\) in the vertical direction, the equation becomes \[ u_t= \frac {u_{xx}}{1+u_x^2} + g(x,u(x))\sqrt{1+u_x^2}. \] By proving some estimates for \(u\) depending only on \(\| g\|_\infty\), the authors obtain an existence and uniqueness result for solutions, when \(g\) is an \(L^\infty\)-function which is independent of \(u\). As application to the homogenization problem, the limit as \(\varepsilon \to 0\) is studied.

In the present paper, the authors classify all possible singularities which can arise during the evolution and show that, when \(g\) is smooth and the initial curve is embedded, the existence time of a regular solution is bounded below by a quantity depending only on \(\| g\|_\infty\) and on the initial curve. Since the authors have no estimates on the curvature in terms of \(\| g\|_\infty\), they are not able to obtain a general existence result for \(g \in L^\infty\). However, by assuming that the initial curve is the graph of a function \(u\) in the vertical direction, the equation becomes \[ u_t= \frac {u_{xx}}{1+u_x^2} + g(x,u(x))\sqrt{1+u_x^2}. \] By proving some estimates for \(u\) depending only on \(\| g\|_\infty\), the authors obtain an existence and uniqueness result for solutions, when \(g\) is an \(L^\infty\)-function which is independent of \(u\). As application to the homogenization problem, the limit as \(\varepsilon \to 0\) is studied.

Reviewer: Shigeru Sakaguchi (Sendai)