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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
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