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• W2891792321 abstract "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." @default.
• W2891792321 created "2018-09-27" @default.
• W2891792321 creator A5076109234 @default.
• W2891792321 date "2018-09-14" @default.
• W2891792321 modified "2023-09-26" @default.
• W2891792321 title "The Curve Diffusion Flow with a Contact Angle" @default.
• W2891792321 hasPublicationYear "2018" @default.
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