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*} (-Δ)^αV- \frac{2α-1}{2α}V+V\cdot\nabla V-\frac{1}{2α}x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from the study of self-similar solutions to the generalized Naiver-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 $H_{\omega }^{\alpha }({\mathbb{R}}^3)$. Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from $H_{\omega }^{\alpha }({\mathbb{R}}^3)$ to $H_{\omega }^{1+\alpha }({\mathbb{R}}^3)$. This regularity result, together linear theory for the non-local Stokes system, lead to pointwise estimates of $V$ which allow us to obtain a natural pointwise property of the self-similar solution constructed in \cite{LXZ}. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Naiver-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai \cite[Comm. Math. Phys., 328 (2014), 29-44]{T}.

47pages, To appear in Trans. AMS

References in corpus (4)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations
• Fine properties of self-similar solutions of the Navier-Stokes equations
• The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system
• Forward self-similar solutions of the fractional Navier-Stokes Equations

Cited by in corpus (2)

• Spatial decay of discretely self-similar solutions to the Navier-Stokes equations
• Global regularity and decay behavior for Leray equations with critical-dissipation and Its Application to Self-similar Solutions