$\epsilon$-regularity criteria in anisotropic Lebesgue spaces and Leray's self-similar solutions to the 3D Navier-Stokes equations

Abstract

In this paper, we establish some $\epsilon$-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier-Stokes equations as follows: $$\underset{\varrho \to 0}{\mathrm{limsup}}{\varrho }^{1-\frac{2}{p}-\sum _{j=1}^3\frac{1}{q_j}}\Vert u{\Vert }_{L_t^p{L}_x^{\overrightarrow{q}}(Q(\varrho ))}\le \epsilon ,\frac{2}{p}+\sum _{j=1}^3\frac{1}{q_j}\le 2\text{with}{q}_j>1;$$$$\underset{-1\le t\le 0}{\sup }\Vert u{\Vert }_{L^{\overrightarrow{q}}(B(1))}<\epsilon ,\frac{1}{q_1}+\frac{1}{q_2}+\frac{1}{q_3}<2\text{with}1<q_j<\infty ;$$ $$\Vert u{\Vert }_{L_t^p{L}_x^{\overrightarrow{q}}(Q(1))}+\Vert \Pi {\Vert }_{L^1(Q(1))}\le \epsilon ,\frac{2}{p}+\sum _{j=1}^3\frac{1}{q_j}<2\text{with}1<q_j<\infty ,$$ which extends the previous results in [2, 12, 18, 19, 22, 37, 43]. As an application, in the spirit of [4], we prove that there does not exist a nontrivial Leray's backward self-similar solution with profiles in $L^{\overrightarrow{p}}({\mathbb{R}}^3)$ with $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}<2$. This generalizes the corresponding results of [4, 20, 28, 38].

There is a mistake in previous version. We removed all the results involving regularity in one direction and added results concerning Leray's backward self-similar in anisotropic Lebesgue spaces