PD Dr. D. Ševčovič (Comenius University Bratislava, Slovak Republic)

Higher order estimates for the curvature and nonlinear stability of stationary solutions for curvature flow with triple junction

We are interested in the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. We derive higher order energy estimates yielding a priori estimates for the H²-norm of the curvature of moving arcs. As a consequence of these estimates we will be able to prove exponential decay of the H²-norm of the curvature. As a consequence, we will show that a linear stability criterion due to Ikota and Yanagida [2] is also sufficient for nonlinear stability of stationary solutions for curvature flow with triple junction. This is a joint work with Harald Garcke and Yoshihito Kohsaka [1].

References:
[1] H. Garcke, Y. Kohsaka and D. Sevcovic: Nonlinear stability of stationary solutions for curvature flow with triple junction, submitted. arXiv:0802.3036
[2] Ikota R. and Yanagida E.: A stability criterion for stationary curves to the curvature-driven motion with a triple junction, Differential Integral Equations 16 (2003), 707--726.