The Curve Diffusion Flow with a Contact Angle
Julia Butz · University of Regensburg Publication Server (University of Regensburg) · 2018
We consider the evolution of open curves driven by curve diffusion flow. This geometric evolution equation arises in problems of phase separation in material science and is the one-dimensional analogue of the surface diffusion flow. The evolving family of curves has free boundary points, which are supported on a line and it has a fixed contact angle $\alpha \in (0, \pi)$ with that line. Moreover, it satisfies a no-flux condition. First, we discuss a result on well-posedness locally in time for curves which can be described by a sufficiently small height function of class $W_2^{\gamma}$, $\gamma \in (\tfrac{3}{2}, 2]$, over a reference curve. In order to proof the result, we reduce the geometric evolution equation to a fourth order quasilinear, parabolic partial differential equation for the height function on a fixed interval. The proof of the well-posedness of this problem is based on a contraction mapping argument: A result on maximal $L_p$-regularity with temporal weights by Meyries and Schnaubelt enables us to solve the linearized problem with optimal regularity. By establishing multiplication results in time weighted anisotropic $L_2$-Sobolev spaces of low regularity, we can to show that the non-linearities are contractive for small times. Furthermore, we show the existence of a suitable reference curve for every admissible initial curve: We smoothen the initial curve by evolving it by a parabolic equation. Afterwards, we establish conditions on the distance of two curves which guarantee that one curve can be used as a reference curve for the other one. By $C_0$-semigroup and interpolation theory, we confirm that the solution of the aforementioned parabolic equation is in fact a viable reference curve. Combining this with the first result, we obtain that the flow starts for every admissible initial curve of class $W_2^{\gamma}$, $\gamma \in (\tfrac{3}{2}, 2]$. By exploiting this result, we can give a blow-up criterion in terms of a $L_2$-bound of the curvature: If a solution of the curve diffusion flow subject to the previously mentioned boundary conditions exists only for a maximal time $T_{max} < \infty$, then the $L_2$-norm of the curvature tends to $\infty$ as $t \rightarrow T_{max}$. For the proof, we assume, contrary to our claim, that the $L_2$-norm of the curvature remains bounded for a sequence in time approaching $T_{\max}$. A compactness argument combined with the short time existence result enables us to extend the flow beyond $T_{max}$, which contradicts the maximality of the solution.