$\varepsilon$-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 $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier-Stokes equations as follows: $$ \limsup\limits_{\varrho\rightarrow0} \varrho^{1-\frac{2}{p}-\sum\limits^{3}_{j=1}\frac{1}{q_{j}}} \|u\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho))} \leq\varepsilon, ~~\frac{2}{p}+\sum\limits^{3}_{j=1}\frac{1}{q_{j}} \leq2~~~~~\text{with}~q_{j} > 1;\\$$$$ \sup_{-1\leq t\leq0}\|u\|_{L^{\overrightarrow{q}}(B(1))} < \varepsilon,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}} <2\quad \text{with}\, 1<q_{j}<\infty;$$ $$\|u \|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(1))} +\|\Pi\|_{L^{1}(Q(1))}\leq\varepsilon, \quad \frac2p+\sum^{3}_{j=1}\frac{1}{q_{j}} <2 ~~~\text{with}~~ 1<q_{j}<\infty, $$ 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].