Joel Dahne: Computer Assisted Studies in Fluid Mechanics and Spectral Geometry

Abstract

This thesis contains four papers in the area of partial differential equations. The first two papers are related to spectral geometry and the last two papers to fluid mechanics. A common theme of the papers is that they make use of computer assisted methods in the proofs.

Paper I concerns the computation of very precise enclosures of eigenvalues of the Laplace-Beltrami operator on spherical triangles. The interest in these eigenvalues comes from a connection with the combinatorial problem of studying discrete random walks.

Paper II gives a concrete counterexample to Payne's nodal line conjecture. The conjecture is concerned with the existence of bounded planar domains for which the second eigenfunction of the Dirichlet Laplacian has a nodal line that doesn't touch the boundary. The paper gives an explicit domain for which it is proved that the nodal line doesn't touch the boundary.

Paper III and IV both prove the existence of a certain type of solution known as a highest cusped traveling wave. Paper III deals with the Burgers-Hilbert equation and Paper IV with a family of fractional Korteweg-de Vries equations. The existence is asserted by constructing an explicit approximation of the solution and proving the existence of an exact solution nearby with the use of a fixed point formulation. The proof not only establishes the existence, but also determines the precise asymptotic behavior of the waves near the cusp.