Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Julia Butz (University of Regensburg)

Short Time Existence for Curve Diffusion Flow for Curves with Boundary Contact

In non-linear PDEs it is always desirable to be able to find strong solutions even if the initial regularity is low, as this gives insights in smoothing properties of the system and helps to provide good blow-up criteria.

In this contribution, we obtain local well-posedness for flexible initial data in $H_2^{4(\mu - \frac{1}{2})}(I)$, $\mu \in (\frac{7}{8}, 1]$ in the case of evolving curves driven by curve diffusion flow. This is done for curves which intersect an external boundary at a fixed angle $\alpha \in (0, \pi)$.

In order to reduce the geometric evolution equation to a quasilinear fourth order PDE on an interval, we use curvilinear coordinates. To establish local well-posedness for this equation and to deal with the rough initial data, we have to work in a setting of temporally weighted $L_p$-spaces. More precisely, we use a linearization procedure and employ a result on maximal $L_p$-regularity with temporal weights [1].

Once we know that the flow starts, a further step is to deduce a blow up rate by adapting the techniques of [2], [3].

References

[1] M. Meyries and R. Schnaubelt. Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions. Mathematische Nachrichten, (285):1032-1051, 2012.
[2] G. Dziuk, E. Kuwert, and R. Schätzle. Evolution of elastic curves in $\mathbb{R}^n$: Existence and computation. SIAM J. Math. Anal., 33(5):1228-1245, 2002.
[3] A. Dall'Acqua and P. Pozzi. A Willmore-Helfrich $L_2$-flow of curves with natural boundary conditions. Comm. Anal. Geom., 22(4):617-669, 2014.