A note on traveling waves for area-preserving geometric flows

Global stability of traveling waves for an area preserving curvature flow with contact angle condition

“…We note that they only study the case ψ ± ∈ (0, π/2) so that the initial curve γ(0) and the profile curve W(0) can be assumed to represent concave graphs. For the case of more general contact angles, the author and Kohsaka [14] studied the existence of traveling waves and geometric properties of these traveling waves under an assumption associated to the winding number − W(0) κ W ds = ψ + + ψ − , where κ W is the curvature of the profile curve W(0). They proved the existence of a traveling wave for any contact angles ψ ± ∈ (0, π).…”

“…We note that the argument for Theorem 1.1 is based on analytic semigroup theory as in [21]. Since the flow is unique, we can discuss the stability of the traveling waves W(t) obtained by [14] by studying the asymptotic behavior of global-in-time solutions to (1.1)-(1.3). In the present paper, we also assume the following properties to study the asymptotic behavior of solutions:…”

“…Assume (A4)-(A7). Let W(t) be the traveling wave obtained by [14] such that A W = A(0). Denote by κ(•, t) and κ * the curvatures of γ(t) and W(0), respectively.…”

“…Let γ(0) satisfies (A'). Denote by γ(t) and W(t) the global-in-time solution to (1.1)-(1.3) obtained by [21] and the traveling wave obtained by [14,21] such that A W = A(0), respectively. Then, κ(•, t) converges exponentially to κ * in C ∞ and there exists a constant a ∈ R such that γ(t) converges to W(t) + a e 1 under the Hausdorff metric, where κ(•, t) and κ * are the curvature of γ(t) and W(0), respectively.…”

“…Some known results are also stated in details to discuss the stability of the traveling waves. The known results relate to the existence theory for traveling waves in [14] and the local stability theory of the traveling waves in [21]. In Section 4, we construct a Lyapunov functional of the curvature and analyze the ω-limit points of the curvature.…”