Abstract
Abstract
Highlights
This study is concerned with the possibility of maintenance of smoothness in Navier– Stokes flows by viscous effects
The requirement of simultaneous blow up of |u| and |p| at a singular point gives a relatively clear picture of Navier–Stokes singularity: high velocity fluid particles crashing upon a global pressure minimum that decreases to negative infinity in a finite time
When the term driving the evolution of ||u||Lq in this equation does not exceed its dissipation counterpart, ||u||Lq does not grow and regularity persists. This “raw” regularity criterion is strongest in the sense that any results derived from it cannot render improvements
Summary
This study is concerned with the possibility of maintenance of smoothness in Navier– Stokes flows by viscous effects. The requirement of simultaneous blow up of |u| and |p| at a singular point gives a relatively clear picture of Navier–Stokes singularity: high velocity fluid particles crashing upon a global pressure minimum (or multiple minima) that decreases to negative infinity in a finite time. With the stringent constraint on singularity development described above in mind, we present and discuss regularity criteria that encapsulate the velocity-pressure correlation as an essential feature. We examine the plausibility of the derived criteria for flow scenarios satisfying the critical scaling of the Navier–Stokes equations and find that singularity may not develop while respecting such scaling. In the usual Cauchy–Schwarz estimate ||χ||L4(B) ≤ ||ψ||1L/24(B) ||φ||1L/42(B), the left-hand side is finite while the right-hand side strongly diverges as ||ψ||L2(B) and ||φ||L4(B) each diverges This illustrates the significance of the correlation between ψ and φ in determining the magnitude of their mixed norms
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
