AN ELASTIC FLOW FOR NONLINEAR SPLINE INTERPOLATIONS IN ℝⁿ

In this paper we use the method of geometric flow on the problem of nonlinear spline interpolations for non-closed curves in n-dimensional Euclidean spaces. The method applies theory of fourth-order parabolic PDEs to each piece of the curve between two successive knot points at which certain dynamic boundary conditions are imposed. We show the existence of global solutions of the elastic flow in suitable Hölder spaces. In the asymptotic limit, as time approaches infinity, solutions subconverge to a stationary solution of the problem. The method of geometric flows provides a new approach for the problem of nonlinear spline interpolations.</p

Travelling fronts for multidimensional nonlinear transport equations

We consider a nonlinear transport equation as a hyperbolic generalisation of the well-known reaction diffusion equation. We show the existence of strictly monotone travelling fronts for the three main types of the nonlinearity: the positive source term, the combustion law, and the bistable case. In the first case there is a whole interval of possible speeds containing its strictly positive minimum. For subtangential nonlinearities we give an explicit expression for the minimal wave speed. 1 Introduction The work of Fisher [7] and Kolmogorov, Petrovsky, Piskounov [11] inspired the study of the asymptotic behaviour of spreading and interacting particles on unbounded domains. Both articles modelled spread and interaction by a reaction-diffusion equations. In particular it was shown that suitable initial configurations converge asymptotically to travelling front solutions. This observation lead Aronson and Weinberger in a series of papers to introduce the concept of the asymptotic speed o..

ON THE GEOMETRIC FLOW OF KIRCHHOFF ELASTIC RODS

elastic rod