Regularity and Critical Thresholds Phenomena in Nonlinear Balance Laws

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The overall goal of this project is the study of critical regularity associated with different nonlinear balance laws. We focus on borderline cases, of interest in various applications, where intrinsic features of the solutions such as smoothness vs. generic finite-time breakdown, time decay, etc., hinge on a delicate balance between nonlinear convection and a variety of (possibly nonlinear) forcing mechanisms. Often this balance is maintained by global invariants of the flow. These include spectral invariants, which in turn lead to critical threshold phenomena, or borderline invariant regularity spaces. A main focal topic of this project is the study of balance laws governed by Eulerian dynamics. Such models show up in different contexts dictated by different forcing. Previous research developed a precise framework for studying the critical threshold phenomena for such Eulerian dynamics. Here we seek extensions to include more realistic models driven by global and non-isotropic forcing. In particular, this project investigates global regularity for multidimensional Euler-Poisson equations with sub-critical initial data and persistence of minimal regularity for Euler and Navier-Stokes equations with initial configurations in borderline regularity spaces.This research project pursues the development of mathematical tools for the understanding of fundamental properties of fluid flow and other dynamical processes. The results will have potential application for the modeling and numerical simulation of a variety of physical phenomena, including the turbulent flow of fluids, that are governed by the type of partial differential equations under study.