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

Abstract:

We investigate a regularity for weak solutions of the following generalized Leray equations \begin{equation*} (-\Delta )^{\alpha }V- \frac {2\alpha -1}{2\alpha }V+V\cdot \nabla V-\frac {1}{2\alpha }x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from the study of self-similar solutions to the generalized Navier-Stokes equations in $\mathbb R^3$. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions $V$ in the weighted Hilbert space $H^\alpha _{\omega }(\mathbb {R}^3)$. Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from $H^\alpha _{\omega }(\mathbb {R}^3)$ to $H_{\omega }^{1+\alpha }(\mathbb {R}^3)$. This regularity result, together with linear theory for the non-local Stokes system, leads to pointwise estimates of $V$ which allow us to obtain a natural pointwise property of the self-similar solution constructed by Lai, Miao, and Zheng [Adv. Math. 352 (2019), pp. 981–1043]. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Navier-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai Comm. Math. Phys., 328 (2014), pp. 29–44.