Curve diffusion and straightening flows on parallel lines

In this paper, we study families of immersed curves $\gamma:(-1,1)\times[0,T)\rightarrow\mathbb{R}^2$ with free boundary supported on parallel lines $\{\eta_1, \eta_2\}:\mathbb{R}\rightarrow\mathbb{R}^2$ evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux condition. We give estimates and monotonicity on the normalised oscillation of curvature, yielding global results for the flows.Comment: 35 pages, 3 figure

On self-similar solutions to the surface diffusion flow equations with contact angle boundary conditions

We consider the surface diffusion flow equation when the curve is given as the graph of a function v(x; t) defined in a half line R+ = {x > 0} under the boundary conditions vx = tan > 0 and vxxx = 0 at x = 0. We construct a unique (spatially bounded) self-similar solution when the angle is sufficiently small.We further prove the stability of this self-similar solution. The problem stems from an equation proposed by W. W. Mullins (1957) to model formation of surface grooves on the grain boundaries, where the second boundary condition vxxx = 0 is replaced by zero slope condition on the curvature of the graph. For construction of a self-similar solution we solves the initial-boundary problem with homogeneous initial data. However, since the problem is quasilinear and initial data is not compatible with the boundary condition a simple application of an abstract theory for quasilinear parabolic equation is not enough for our purpose. We use a semi-divergence structure to construct a solution. 2010 Mathematics Subject Classification: Primary 35C06; Secondary 35G31, 35K59, 74N20. Keywords: Self-similar solution; Surface diffusion flow; Stability; Analytic semigroup; Mild solution

Short Time Existence for the Curve Diffusion Flow with a Contact Angle

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle $\alpha \in (0, \pi)$: The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class $W_2^{\gamma}$ with $\gamma \in (\tfrac{3}{2}, 2]$. For the proof the evolving curve is represented by a height function over a reference curve: The local well-posedness of the resulting quasilinear, parabolic, fourth-order PDE for the height function is proven with the help of contraction mapping principle. Difficulties arise due to the low regularity of the initial curve. To this end, we have to establish suitable product estimates in time weighted anisotropic $L_2$-Sobolev spaces of low regularity for proving that the non-linearities are well-defined and contractive for small times.Comment: 38 page

Nonlinear stability of stationary solutions for surface diffusion with boundary conditions

The volume preserving fourth order surface diffusion flow has constant mean curvature hypersurfaces as stationary solutions. We show nonlinear stability of certain stationary curves in the plane which meet an exterior boundary with a prescribed contact angle. Methods include semigroup theory, energy arguments, geometric analysis and variational calculus