Masters´ Thesis Defense by Benjamin Halager Andersen

Title: Application of Wavelets to Nonlinear Partial Differential Equations and The Phenomenon of Turbulence

Abstract: Wavelet techniques has long been successfully applied to the study of differential equations. The present thesis explore some of these techniques and develop them further. Especially, heatlets (i.e. wavelet fundamental solutions to the heat equation) is used to construct a new family of exact solutions to the viscous Burgers’ equation, on a periodic domain subject to spatially vanishing Dirichlet boundary conditions. A demonstration of this construction is done using the Haar system.
A new numerical scheme for the viscous Burgers’ equation based on the Haar-wavelets is developed. The scheme is compared to a traditional spectral scheme in the case of a sinusoidal initial conditions. The temporal evolution of the L2- and L1-error is computed at various resolutions. It become clear that the Haar-wavelet scheme is incapable to compete with the naïve spectral scheme.
Starting from the divergence-free biorthogonal vector wavelet decomposition of velocity field, a Obukhov-like shell model with nearest neighbour interactions and geometrically separated shells is derived. The model is falsely presumed to approximate the two-dimensional Navier-Stokes equations. It is proven that the model posses a unique unstable inviscid fixed point in the limit N ! 1. However, the fixed point fails to reproduce the correct energy spectrum corresponding to two-dimensional turbulence, nor does the model fulfil Liouville’s theorem. Lastly and unrelated to the theory wavelets, we prove that the PT -symmetric extension of the advection equation posses exact solutions that does not embody shock formation for ✏ 2 R \ 2Z.