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

Deuring, Paul

Math. Bohem. 138, No. 3, 299-324 (2013).

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.

Reviewer: Bernard Ducomet (Bruyères le Châtel)

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