The Cauchy problem for the homogeneous time-dependent Oseen system in $ \mathbb {R}^3 $: spatial decay of the velocity

Deuring, Paul

Mathematica Bohemica, Tome 138 (2013) no. 3, pp. 299-324.

Résumé

We consider the homogeneous time-dependent Oseen system in the whole space $ \mathbb {R}^3 $. The initial data is assumed to behave as $O(|x|^{-1- \epsilon })$, and its gradient as $O(|x|^{-3/2- \epsilon })$, when $|x|$ tends to infinity, where $\epsilon $ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $|u(x,t)|=O(|x|^{-1})$, and its spatial gradient $\nabla _xu$ decreases with the rate $|x|^{-3/2}$, for $|x|$ tending to infinity, uniformly with respect to the time variable $t$. Since these decay rates are optimal even in the stationary case, they should also be the best possible in the evolutionary case considered in this article. We also treat the case $\epsilon =0$. Then the preceding decay rates of $u$ remain valid, but they are no longer uniform with respect to $t$.

Détail

DOI : 10.21136/MB.2013.143439

Classification : 35B25, 35Q30, 35Q35, 65N30, 76D05
Keywords: Cauchy problem; time-dependent Oseen system; spatial decay; wake