ABSTRACT
The problem of the regularity of solutions of the Navier-Stokes equations is related to a number of physical and mathematical concepts. A nonexhaustive list includes vorticity, filtering and renormalization, Ho¨lder exponents, Besov spaces, intermittent cascades, Euler dissipation, scaling, ladder inequalities, multifractals, and fractional derivatives. The class of Hermitian continuous wavelet transforms overlaps with many of these topics, and seems well suited to study NS regularity. The asymptotic scaling of the wavelet coefficients is related to the local Ho¨lder exponent h. An anisotropic version of the Ho¨lder exponent is proposed, and the exact evolution equation for the wavelet coefficients is derived. The small-scale asymptotics of the various terms can be estimated, and the dominant exponents evaluated. A runaway singularity corresponds to the smallest exponent associated with the nonlinear terms, which would then dominate the evolution. In the case of isotropic Ho¨lder exponents, a runaway singularity would occur for h < 1/3 for three-dimensional Euler turbulence, but the viscous term would maintain regularity. No firm conclusion is reached yet about anisotropic exponents, but preliminary results do not show evidence of a sharp cross-over value of h.
