The problem of whether a singularity can form in a flow described by the 3D Navier-Stokes equations is one of the major open problems in mathematical physics. Its significance stems from the fact that the Navier-Stokes equations have been widely used in science and engineering to model 3D fluid flows, and a rigorous validation of the model in the regime of the intense fluid activity (e.g., the pockets of turbulence encountered by the airplanes), and in particular ruling out the singularity formation, is the key to more reliable modeling of turbulent flows. Since the inception of the Navier-Stokes regularity problem in the 1930s, there has been a 'gap' between what is needed to prevent the possible formation of a singularity and what could be rigorously obtained from the equations. The aim of the research to be accomplished under the award is to bridge this gap and arrive at 'criticality'; this will not completely rule out the formation of singularities, but will drastically restrict the possible scenarios at which it might occur. Given the omnipresence of the turbulent phenomena, both in nature and in the engineered world, the impact of the research will extend beyond the boundaries of the discipline. In the domain of mentoring junior scientists, the award will support a graduate research assistant.

The research to be carried out under the award builds on a very recent work by the PI and L. Xu demonstrating--for the first time--asymptotically critical nature of the Navier-Stokes regularity problem. The methodology is based on the study of a suitably defined scale of sparseness of the super-level sets of the positive and the negative parts of the components of the higher-order derivatives of the velocity field, and the sparseness is utilized via the harmonic measure majorization principle (the positive and the negative parts of the components are subharmonic since any smooth flow is automatically analytic in the spatial variables). The sparser the super-level sets are, the more efficient the harmonic measure majorization principle is in generating the 'self-improving' bounds on the sup-norm. Within this framework, it was demonstrated that the 'scaling gap' between the regularity class and the corresponding a priori bound shrinks to zero as the order of the derivative goes to infinity, a manifestation of the asymptotic criticality. The main goal of the research to be performed is to arrive at stronger and/or more classical manifestations of the criticality. There are two natural avenues to take, toward criticality with respect to the strength of the diffusion and toward criticality with respect to the strength of the nonlinearity. These are exactly the two main projects, the former will take place in the setting of the 3D,super-critical, hyper-dissipative Navier-Stokes equations, and the latter in the realm of the geometric depletion of the nonlinearity via the local coherence of the vorticity direction.

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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Pedro Embid
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University of Virginia
United States
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