Global Regularity of weak solutions to the generalized Leray equations and its applications

Abstract

We investigate the regularity for weak solutions of the following generalized Leray equationswhich arises from the study of similar solutions to the generalized Naiver-Stokes equations in R 3 . Firstly, by making use of the vanishing viscosity and developing the non-local effect of the fractional diffusion operator, we give the unifrom estimate for weak solution V in the weightedHilbert space H α ω (R 3 ). Via the differences characterization of Besov spaces and the bootatrap argument, we improve the regularity for weak solution from H α ω (R 3 ) to H 1+α ω (R 3 ). This regularity result, together linear theory for the non-local stokes system, deduces the pointwise estimates of V which makes us to obtain the natural pointwise property of the self-similar solution constructed in [12]. In particular, we obtain the optimal decay estimate to the self-similar solution of the classical Naiver-Stokes equations by means of the special structure of Oseen tensor under the low regularity condition on initial data. This answers the question proposed by Tsai in [28].