Session S37 - New Developments in Mathematical Fluid Dynamics

Monday, July 19, 20:00 ~ 20:25 UTC-3

On criticality of the Navier-Stokes diffusion

Zoran Grujić

University of Virginia, USA

It has been known since the work of J.-L. Lions in 1960s that the hyper-dissipative (HD) Navier-Stokes (NS) system is regular as long as the diffusion exponent beta is greater or equal to 5/4 (5/4 is critical in the sense that the unique scaling-invariance of the system takes place at the energy level).

The goal of this talk is to present a mathematical framework--based on the scale of sparseness of the super-level sets of the higher-order derivatives--in which a HD NS flow near a potential spatiotemporal singularity is classified in three categories: `turbulent' (higher-order derivatives are dominant), `steady' (derivatives of different orders are comparable), and `laminar' (lower-order derivatives are dominant). In the laminar scenario, the blow-up is ruled out for any beta greater or equal to one, with no assumptions. In the turbulent scenario, the blow-up is ruled out for any beta greater than one, with no assumptions. In the steady scenario, the blow-up is ruled out for any beta greater than one, with an assumption that the flow exhibits a certain `focusing' property in the vicinity of the potential singularity. This is a joint work with Liaosha Xu.

Joint work with Liaosha Xu (University of Virginia, USA).

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