Three problems in fluid mechanics through the lens of sparseness of the regions of intense fluid activity

Turbulent flows are omnipresent, from the wake turbulence behind an airliner to the solar wind turbulence. Notwithstanding significant progress, a complete understanding of turbulent motion remains one of the grand challenges in science and engineering. In particular, a precise description of the role the coherent vortex structures (e.g., the wingtip vortices or the current sheets in the solar wind) play in turbulence remains elusive. The fact that the regions of the intense fluid activity self-organize in these coherent structures is also referred to as the 'spatial intermittency' of turbulent flows. The goal of the project is to utilize a novel mathematical framework for the study of spatial intermittency in fluid flows introduced by the principal investigator (PI) to tackle several open problems in the field, including the classical problem of the vortex sheet roll-up. The project will also provide opportunities for the involvement of the students in the research. The overarching objective of this project is to demonstrate the broader utility of the mathematical framework based on the spatial intermittency of the regions of intense fluid activity that had been developed by the PI and the collaborators in the quest to bridge the scaling gap in the Navier-Stokes (NS) regularity problem. The project branches into three subprojects. The first one is in the context of hyper-diffusion. Here, the goal is to show that–as soon as the power of the Laplacian is strictly greater than one–locality of the nonlinear interactions will imply regularity. In addition to the mathematical interest, this is also intriguing from the standpoint of the physics of turbulence since the locality of the nonlinear transfer is one of the tenets of turbulence phenomenology. The second one will consider the NS flows in several critical spaces in which only the small data regularity results are known. The key idea here is to gain a logarithm (in the case of large data) by reformulating the concept of local one-dimensional sparseness at scale in terms of the local directional maximal function. The third one concerns the classical problem of the vortex sheet roll-up. The main goal here is to investigate a stabilizing effect of the viscosity–considering the full NS system–based on local one-dimensional sparseness of the vorticity super-level sets. More precisely, the framework will naturally identify a family of constraints among viscosity, on one side, and the thickness and the curvature of the sheet on the other. 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.