Short time existence for the curve diffusion flow with a contact angle

Abstract

We prove a blow-up criterion in terms of an L 2 -bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle α ∈ (0, π) with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.

Global solution and global orbit to reaction–diffusion equation for fractional Dirichlet‐to‐Neumann operator with subcritical exponent

“…Therefore u ∈ W 2s,1 q (Q ∞ ) = L q ([t 0 , ∞); W 2s q (Ω × {0})) ⋂ W 1 q ([t 0 , ∞); L q (Ω × {0})) for any 1 ≤ q < ∞, where Q ∞ = Ω×{0}×[t 0 , ∞), and by the embedding of anisotropic spaces, 22,23 it follows that u ∈ BUC([t 0 , ∞); BUC (Ω × {0})), where 0 < ≤ 2s − N + 2s/q, and BUC (Ω × {0}) is the Banach space of bounded Hölder continuous functions of order on Ω × {0} for ∈ R + ∖N, while for ∈ N 0 , BUC (Ω × {0}) is the Banach space of -times bounded uniformly continuously differentiable functions on Ω × {0} (see Meyries 23 A.4). Furthermore, applying the standard bootstrap argument, 24 we finally obtain u(x, t) is a classical solution for all t ≥ t 0 > 0, which completes the proof of Theorem 1.4.…”

A Survey of the Elastic Flow of Curves and Networks

“…It is worth to mention also results about the Helfrich flow [17,56], the elastic flow with constraints [25,43,44] and other fourth (or higher) order flows [1,2,36,37,57].…”

On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

“…[AGM16]. In [AB18], Abels and Butz proved analytic existence for the evolution of open curves with respect to curve diffusion flow, which is the one dimensional pendant to the surface diffusion flow, with a contact angle and rough initial data. Indeed, for curves there are already methods to prove geometric existence and uniqueness.…”