Critical Regularity, Selection Dynamics, and Condensation in Nonlinear Balance Laws

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In many mathematical models of physical reality, persistent patterns are formed and maintained by a balance of competing influences. Such phenomena arise naturally in fluid dynamics, evolutionary biology, quantum mechanics, and others. Examples range from forming eddies and coherent structures in turbulent flows to shock formation in gases, or natural selection in population dynamics, or formation of Bose-Einstein condensate (a new state of matter) in quantum mechanics. The goal of the current project is to develop novel mathematical tools and numerical algorithms for analyzing how such competing effects achieve dynamic balance or lead to critical solution behavior. The mathematical results are expected to be fundamental, and contribute to a body of understanding that promises to be useful to researchers across a range of disciplines. Critical regularity is a fundamental problem in fluid dynamics, the study of which can provide a deeper understanding of critical threshold phenomena occurring in wider applications. The understanding of photon condensate can result in methods which may potentially be suitable for designing novel light sources. This investigation focuses on the study of dynamic behavior in areas strongly motivated by applications and the theory of partial differential equations, with research objectives ranging from critical regularity in nonlinear balance laws, selection dynamics in trait-structured population models to energy transport in photon scattering. The mathematical models have one striking feature in common: the underlying dissipation is insufficient to prevent finite time singularity formation, and the persistence of the dynamic behavior hinges on a delicate balance among competing forces. These solution features depend on crossing a critical threshold associated with the initial configuration and/or the interaction kernels. High order numerical algorithms will be developed to solve these problems, with a unified approach so that several intrinsic solution structures are retained at the discrete level. Structure-preserving algorithms as such are important in capturing the correct physics over long time simulations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.