Spatial Decay of the Vorticity Field of Time-Periodic Viscous Flow Past a Body

We study the asymptotic spatial behavior of the vorticity field, omega(x, t), associated to a time-periodic Navier-Stokes flow past a body, B, in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, R, omega decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by (omega) over bar its time-average over a period and by omega P := omega - (omega) over bar its purely periodic component, we prove that inside R, (omega) over bar 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 B, omega(x, t) behaves like the vorticity field of the corresponding steady-state problem.