We study the asymptotic spatial behavior of the vorticity field, \(\omega (x,t)\), associated to a time-periodic Navier–Stokes flow past a body, \({\mathscr {B}}\), in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, \({\mathcal {R}}\), \(\omega \) decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by \({\bar{\omega }}\) its time-average over a period and by \(\omega _P:=\omega -{\bar{\omega }}\) its purely periodic component, we prove that inside \({\mathcal {R}}\), \({\bar{\omega }}\) has the same algebraic decay as that known for the associated steady-state problem, whereas \(\omega _P\) decays even faster, uniformly in time. This implies, in particular, that “sufficiently far” from \({\mathscr {B}}\), \(\omega (x,t)\) behaves like the vorticity field of the corresponding steady-state problem.

我们研究了涡度场\(\omega (x,t)\)的渐近空间行为，与经过身体的时间周期纳维-斯托克斯流相关，\({\mathscr {B}}\)，在满足类 Serrin 条件的一类弱解。我们表明，在尾流区域之外，\({\mathcal {R}}\)，\(\omega \)以指数速率逐点衰减，在时间上均匀。此外，用\({\bar{\omega }}\)表示一段时间内的时间平均值和\(\omega _P:=\omega -{\bar{\omega }}\) 表示它的纯周期分量，我们证明在\({\mathcal {R}}\) , \({\bar{\omega }}\)具有与已知的相关稳态问题相同的代数衰减，而\(\omega _P\)衰减得更快，时间均匀。这特别意味着，离\({\mathscr {B}}\) “足够远” ，\(\omega (x,t)\)表现得像相应稳态问题的涡量场。