For any discretely self-similar, incompressible initial data v 0 which satisfies v 0 L 3 w (R 3 ) ≤ c 0 where c 0 is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemarié-Rieusset to the 3D Navier-Stokes equations in the whole space. No further assumptions are imposed on the initial data; in particular, the data is not required to be continuous or locally bounded on R 3 \ {0}. The same method gives a third construction of self-similar solutions (after those in [9] and [14]) and works for any −1-homogeneous initial data in L 3 w .

It is shown-within a mathematical framework based on the suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the vorticity components, and in the context of a blow-up-type argument-that the ever-resisting 'scaling gap', i.e., the scaling distance between a regularity criterion and a corresponding a priori bound (shortly, a measure of the super-criticality of the 3D Navier-Stokes regularity problem), can be reduced by an algebraic factor; since (independent) fundamental works of Ladyzhenskaya, Prodi and Serrin as well as Kato and Fujita in 1960s, all the reductions have been logarithmic in nature, regardless of the functional set up utilized. More precisely, it is shown that it is possible to obtain an a priori bound that is algebraically better than the energy-level bound, while keeping the corresponding regularity criterion at the same level as all the classical regularity criteria. The mathematics presented was inspired by morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of turbulent flows, as well as by the physics of turbulent cascades and turbulent dissipation.

Chae and Wolf recently constructed discretely self-similar solutions to the Navier-Stokes equations for any discretely self similar data in L 2 loc . Their solutions are in the class of local Leray solutions with projected pressure, and satisfy the "local energy inequality with projected pressure". In this note, for the same class of initial data, we construct discretely self-similar suitable weak solutions to the Navier-Stokes equations that satisfy the classical local energy inequality of Scheffer and Caffarelli-Kohn-Nirenberg. We also obtain an explicit formula for the pressure in terms of the velocity. Our argument involves a new purely local energy estimate for discretely self-similar solutions with data in L 2 loc and an approximation of divergence free, discretely self-similar vector fields in L 2 loc by divergence free, discretely self-similar elements of L 3 w .

We introduce new classes of solutions to the three dimensional Navier-Stokes equations in the whole and half spaces that add rotational correction to selfsimilar and discretely self-similar solutions. We construct forward solutions in these new classes for arbitrarily large initial data in L 3 w on the whole and half spaces. We also comment on the backward case.

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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.

This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L 2based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical L 2 -based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.1. If u 0 ∈ M 2,r where 0 ≤ r < 1, then u has eventual regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently large t. If u 0 ∈ M 2,r and 1 < r ≤ 3, then u has initial regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently small t. If u, then u has initial and eventual regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently small and large t. In particular, this is true if u 0 ∈ L 3,q for 1 ≤ q < ∞.

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