A CONJECTURE REGARDING LOCAL BEHAVIOR OF VORTICITY IN THE 3D INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Using the 3d incompressible Navier-Stokes equations, it is conjectured how the L(2)-norms of the vorticity omega and its derivatives within a local volume of fluid may be dynamically related to, and finitely co ntrolled by, multifractal properties of this volume. To achieve this, the idea of time dependent average ''stretching'' and ''contracting'' exponents, respectively s(1) (t) and s(2)(t), is introduced, where s(1 ) + s(2) = 3. When natural length scales related to omega acid its der ivatives are rapidly decreasing in a time interval, it is conjectured how a fluid volume may deform into a set of spatial ''dimension'' s = s(1) - s(2), where s satisfies s < 0 in this interval. A plausibility argument produces the uniform bound of s(1) < 2.5 and s(2) > 0.5, indi cating that the final state is a quasi two-dimensional pancake-like ob ject which possesses a fractal-like structure supported by many length scales.