Explorations of singularities in some hydrodynamical partial differential equations

Abstract

The global regularity problem for the 3D Euler equations remains unsolved. The presence of finite-time singularities (FTS) can signal the breakdown of the PDE as a physical model for fluid flow; it is also conjectured to be important for understanding fluid turbulence. Numerical investigations of this problem, albeit painstaking, are useful to isolate candidate initial conditions for FTS. However, the evidence for/against FTS is obscure at best since most numerical schemes breakdown near the time of singularity, due to finite-precision and finite-resolution artefacts. Over the past decades, powerful and creative numerical schemes have emerged from the pursuit of this delicate problem. Owing to the high sensitivity of FTS to errors and numerical artefacts, this problem has catalysed research in the areas of numerical analysis, parallel computing and fluid dynamics alike. The work done in this thesis adds on to the existing body of research in this very direction.