ABSTRACT

It is shown that a local-in-time strong solution u to the 3D Navier–Stokes equations remains regular on an interval $(0,T)$ provided a smallness ${ϵ}_0$-condition on u in a dynamically restricted local Morrey space is stipulated; more precisely, $\underset{t\in (0,T)}{sup}\underset{x\in {\mathbb{R}}^3,\eta (t)\le r\le 1}{sup}\frac{1}{r^{\alpha }}{\int }_{B_r(x)}|u(y,t){|}^p\text{d}y\le {ϵ}_0,$where η is a dynamic dissipation scale consistent with the turbulence phenomenology and α and p are suitable parameters. Such regularity criterion guarantees the volumetric sparseness of local spatial structure of intense vorticity components, preventing the formation of the finite-time blow up at T under the framework of $Z_{\alpha }$-sparseness classes introduced in Bradshaw et al. (An algebraic reduction of the ‘Scaling Gap’ in the Navier–Stokes regularity problem. Arch Ration Mech Anal. 2019;231(3):1983–2005).

摘要

结果表明，3D Navier-Stokes 方程的局部时间强解u在区间上保持规则$(0,吨)$提供了一个小${\epsilon }_0$-在动态受限的局部莫雷空间中规定了u的条件；更确切地说，$\underset{吨\in (0,吨)}{支持}\underset{X\in {\mathbb{R}}^3,\eta (吨)\le r\le 1}{支持}\frac{1}{r^{\alpha }}{\int }_{{乙}_r(X)}|你(\mathrm{是的},吨){|}^p\text{d}\mathrm{是的}\le {\epsilon }_0,$其中η是与湍流现象学一致的动态耗散尺度，α和p是合适的参数。这样的规律性判据保证了强涡度分量的局部空间结构的体积稀疏性，防止了在$Z_{\alpha }$- Bradshaw 等人引入的稀疏类。（Navier-Stokes 规律性问题中“缩放差距”的代数减少。Arch Ration Mech Anal. 2019;231(3):1983-2005）。