Introduction Program Talks & posters Participants Practical Info
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.
News
Stefan Schwede is invited speaker at the ECM 2024 in Sevilla
Jessica Fintzen wins Cole Prize
Regula Krapf receives university teaching award
Catharina Stroppel joined the North Rhine-Westphalia Academy for Sciences and Arts
Daniel Huybrechts receives the Compositio Prize for the periode 2017-2019
Catharina Stroppel receives Gottfried Wilhelm Leibniz Prize 2023
Grants for Mathematics students from Ukraine
Jessica Fintzen is awarded a Whitehead Prize of the London Mathematical Society
Peter Scholze elected as Foreign Member of the Royal Society