My PhD work has focused on the analysis of a large family of PDEs called the EPDiff equation. Interpreting this family as the geodesic equation of a right-invariant Riemannian metric on a diffeomorphism group has allowed us to obtain results on the breakdown of solutions and hence on global well-posedness for the EPDiff equation. We developed a novel comparison theory to obtain a breakdown criterion for these equations and have used it to obtain new breakdown results for the EPDiff family, getting us closer to a complete characterization of global well-posedness for EPDiff and geodesic completeness for the corresponding diffeomorphism group. We have also obtained a local well-posedness result for the higher-dimensional b-equation, using the classic argument of Ebin and Marsden (1970) whereby one writes the PDE in question as an ODE on a Banach manifold, applies Picard iteration to obtain local well-posedness, and transfers the result to the smooth category by showing that there is no loss of spatial regularity during the time-evolution of the ODE.
In Preparation:
1. M. Bauer, S.C. Preston, and J. Valletta. “Breakdown of smooth solutions of the higher-order EPDiff equation with homogeneous Sobolev inertia operator”.
Preprint available:
2. J. Valletta "The b-equation on R^n: a geometric perspective." arXiv:2501.03551
Published:
3. M. Bauer, S.C. Preston, and J. Valletta. “Liouville comparison theory for breakdown of Euler-Arnold equations”. Journal of Differential Equations 407 (2024), pp. 392–431. issn: 0022-0396. doi: https://doi.org/10.1016/j.jde.2024.07.009.
arXiv:2501.03551