MenuEvents for 03/09/2021 from all calendars
Nonlinear Partial Differential Equations
Time: 3:00PM - 4:00PM
Location: Zoom
Speaker: Zoran Grujic, University of Virginia
Title: Global-in-time regularity of the hyper-dissipative Navier-Stokes system - turbulent scenario
Abstract: The question of whether a solution to the 3D Navier-Stokes (NS) system can exhibit a spontaneous formation of singularities, the so-called NS regularity problem, is a fundamental open problem in mathematical physics. It is also intimately related to the phenomena of turbulent cascades and turbulent dissipation, the key tenets of turbulence phenomenology. A major obstacle to progress has been the super-criticality of the system, more precisely, there is a `scaling gap' between the energy level and the level at which the linear (diffusion) and the nonlinear (transport) terms--at least formally--equilibrate. This scaling gap is perhaps best illustrated in the realm of the hyper-dissipative NS system. Here, the NS dissipation (generated by the Laplacian) is replaced by the hyper-dissipation (generated by a fractional power of the Laplacian, say \beta, greater than one). In this setting, the intrinsic scaling of the system indicates that any hyper-dissipation with \beta greater or equal to 5/4 should win over the nonlinearity and prevent a possible formation of singularities. Indeed, this was rigorously established in the foundational work of J.L. Lions in 1960s. In contrast, what happens in the super-critical regime, \beta less than 5/4, remained a mystery. The purpose of this talk is to review a very recent work by Grujic and Xu establishing that as long as \beta is greater than 1 and the flow is in a (suitably defined) turbulent regime, no singularity can form. More precisely (whenever \beta is greater than 1), smallness of a higher-order analogue of the Taylor micro-scale triggers the higher-order cascade which then continues down to the higher-order dissipation scale, preventing any singularity formation. Recall that smallness of the Taylor micro-scale is an indicator of turbulence in the classical phenomenology.
