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F@Willmore geometric flow by a kinetic approach

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Wilmore flow is gradient flow of surfaces evolving so they tend to minimise
the Willmore functional. $W=\int H^2 dS$ where $H$ is the mean curvature of
the evolving surface and $dS$ is the surface area measure. The functional $W$
describes bending energy of an elastic membrane. The functional $W$ was a subject
of great interest in geometry and has applications in biophysics of lipid membranes.
The velocity of the surface in Willmore flow is proportional to $\Delta H +2H(H^2-K)$
where $K$ is the Gauss curvature and $\Delta$ is the Laplace-Beltrami operator on the
surface that is a genuine nonlinear PDE of the forth order for the position of the surface.

We intoduce an approach to the Willmore flow and a family of other geometric flows
based on considering sharp fronts in solutions to kinetic equations for an artificial
gas of particles with chemical reactions. It implies a simple convolution-thresholding
dynamics that by chosing apropriate parameters lets to approximate a family of geometric
flows including generalised curvature flows and the Willmore flow. The mathematical
background and the numerical illustrations with applications of this approach will be presented.

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