## The Cauchy problem for the homogeneous time-dependent Oseen system in $$\mathbb {R}^3$$: spatial decay of the velocity.(English)Zbl 1289.35244

In this article the author considers the Cauchy problem for the homogeneous Oseen system in the whole space $$\mathbb {R}^3.$$
Assuming, for $$\varepsilon >0$$ small enough, a $$O(| x| ^{-1-\varepsilon })$$ decay for the initial velocity $$\vec u_0$$ and a $$O(| x| ^{-3/2-\varepsilon })$$ decay for its gradient, for large $$| x|$$, he proves that $$\vec u$$ (resp. its gradient) actually decays as $$O(| x| ^{-1})$$ (resp. as $$O(| x| ^{-3/2}))$$, uniformly in time.
When $$\varepsilon =0$$, he finally shows that the same decays hold but without uniformity in time.

### MSC:

 35Q30 Navier-Stokes equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids

### Keywords:

Cauchy problem; time-dependent Oseen system; spatial decay; wake
Full Text: