Zachary Bradshaw

Rotationally corrected scaling invariant solutions to the Navier–Stokes equations

ABSTRACT We introduce new classes of solutions to the three-dimensional Navier–Stokes equations in the whole and half-spaces that add rotational correction to self-similar and discretely self-similar solutions. We construct forward solutions in these new classes for arbitrarily large initial data in on the whole and half-spaces. As a special case, this gives a construction of self-similar and discretely self-similar solutions on the half-space. We also comment on the backward case.

Blow-Up Scenarios for the 3D Navier–Stokes Equations Exhibiting Sub-Criticality with Respect to the Scaling of One-Dimensional Local Sparseness

It is shown that, if the vorticity magnitude associated with a (presumed singular) three-dimensional incompressible Navier–Stokes flow blows-up in a manner exhibiting certain time dependent local structure, then time independent bounds on the L1 norm of

Frequency Localized Regularity Criteria for the 3D Navier–Stokes Equations

{|\omega| \log \sqrt{1+ |\omega|^2}}

Minimum cycle bases of direct products of graphs with cycles

|ω|log1+|ω|2 follow. The implication is that the volume of the region of high vorticity decays at a rate of greater order than a rate connected to the critical scaling of one-dimensional local sparseness and, consequently, the solution becomes sub-critical.

Local analyticity radii of solutions to the 3D Navier–Stokes equations with locally analytic forcing

Working directly from the 3D magnetohydrodynamical equations and entirely in physical scales we formulate a scenario wherein the enstrophy flux exhibits cascade-like properties. In particular we show the inertially driven transport of current and vorticity enstrophy is from larger to smaller scale structures and this inter-scale transfer is local and occurs at a nearly constant rate. This process is reminiscent of the direct cascades exhibited by certain ideal invariants in turbulent plasmas. Our results are consistent with the physically and numerically supported picture that current and vorticity concentrate on small-scale, coherent structures.

Minimum cycle bases for direct products of K 2 with complete graphs.

Two regularity criteria are established to highlight which Littlewood–Paley frequencies play an essential role in possible singularity formation in a Leray–Hopf weak solution to the Navier–Stokes equations in three spatial dimensions. One of these is a frequency localized refinement of known Ladyzhenskaya–Prodi–Serrin-type regularity criteria restricted to a finite window of frequencies, the lower bound of which diverges to

L \\log L

{+\infty}