Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2020-09-22 , DOI: 10.1007/s00033-020-01400-x Yanqing Wang , Gang Wu , Daoguo Zhou
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:
$$\begin{aligned}&\limsup \limits _{\varrho \rightarrow 0} \varrho ^{1-\frac{2}{p}-\sum \limits ^{3}_{j=1}\frac{1}{q_{j}}} \Vert u\Vert _{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho ))} \le \varepsilon , ~~\frac{2}{p}+\sum \limits ^{3}_{j=1}\frac{1}{q_{j}} {\le 2}~~~~~\text {with}~q_{j} > 1; \\&\quad \sup _{-1\le t\le 0}\Vert u\Vert _{L^{\overrightarrow{q}}(B(1))} \le \varepsilon ,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}}<2\quad \text {with}\, 1<q_{j}<\infty ; \\&\Vert u \Vert _{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(1))} +\Vert \Pi \Vert _{L^{1}(Q(1))}\le \varepsilon , \quad \frac{2}{p}+\sum ^{3}_{j=1}\frac{1}{q_{j}}<2 ~~~\text {with}~~ 1<q_{j}<\infty , \end{aligned}$$(0.1)which extends the previous results in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982), Choi and Vasseur (Ann Inst H Poincaré Anal Non Linéaire 31:899–945, 2014), Gustafson et al. (Commun Math Phys 273:161–176, 2007), Guevara and Phuc Calc Var 56:68, 2017), He et al. (J Nonlinear Sci 29:2681–2698, 2019), Tian and Xin (Commun Anal Geom 7:221–257, 1999) and Wolf (Ann Univ Ferrara 61:149–171, 2015). As an application, in the spirit of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), 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 Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), Guevara and Phuc (SIAM J Math Anal 50:541–556, 2017), Nečas et al. (Acta Math 176, 283–294, 1996) and Tsai (Arch Ration Mech Anal 143(1):29–51, 1998).
中文翻译：
$$ \ varepsilon $$ε-各向异性Lebesgue空间中的正则性准则和Leray对3D Navier–Stokes方程的自相似解
在本文中，我们为3D Navier–Stokes方程的合适弱解，在各向异性Lebesgue空间中建立了一些\（\ varepsilon \）正则性准则，如下所示：
$$ \ begin {aligned}＆\ limsup \ limits _ {\ varrho \ rightarrow 0} \ varrho ^ {1- \ frac {2} {p}-\ sum \ limits ^ {3} _ {j = 1} \ frac {1} {q_ {j}}} \ Vert u \ Vert _ {L_ {t} ^ {p} L ^ {\ overrightarrow {q}} _ {x}（Q（\ varrho））} \ le \ varepsilon，~~ \ frac {2} {p} + \ sum \ limits ^ {3} _ {j = 1} \ frac {1} {q_ {j}} {\ le 2} ~~~~~ \ text {with}〜q_ {j}> 1; \\＆\ quad \ sup _ {-1 \ le t \ le 0} \ Vert u \ Vert _ {L ^ {\ overrightarrow {q}}（B（1））} \ le \ varepsilon，~~ \ frac {1} {q_ {1}} + \ frac {1} {q_ {2}} + \ frac {1} {q_ {3}} <2 \ quad \ text {with} \，1 <q_ {j} <\ infty; \\＆\验证u \验证_ {L_ {t} ^ {p} L ^ {\ overrightarrow {q}} _ {x}（Q（1））} + \验证\ Pi \验证_ {L ^ { 1}（Q（1））} \ le \ varepsilon，\ quad \ frac {2} {p} + \ sum ^ {3} _ {j = 1} \ frac {1} {q_ {j}} <2 ~~~ \ text {with} ~~ 1 <q_ {j} <\ infty，\ end {aligned} $$（0.1）这扩展了Caffarelli等人先前的结果。（Commun Pure Appl Math 35：771-831，1982），Choi和Vasseur（Ann Inst HPoincaréAnal NonLinéaire31：899-945，2014），Gustafson等。He等（Commun Math Phys 273：161–176，2007），Guevara和Phuc Calc Var 56:68，2017）。（J Nonlinear Sci 29：2681–2698，2019），Tian and Xin（Commun Anal Geom 7：221–257，1999）和Wolf（Ann Univ Ferrara 61：149–171，2015）。作为一种应用，本着Chae和Wolf的精神（Arch Ration Mech Anal 225：549–572，2017），我们证明了在（（L ^ {\ overrightarrow）中，不存在轮廓不无关紧要的Leray向后自相似解{p}}（\ mathbb {R} ^ {3}）\）与\（\ frac {1} {p_ {1}} + \ frac {1} {p_ {2}} + \ frac {1} { p_ {3}} <2 \）。这概括了Chae和Wolf（Arch Ration Mech Anal 225：549-572，2017），Guevara和Phuc（SIAM J Math Anal 50：541-556，2017），Nečas等人的相应结果。（Acta Math 176，283–294，1996）和Tsai（Arch Ration Mech Anal 143（1）：29–51，1998）。
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