On the Time Decay of the Solutions of the Navier–Stokes System

Abstract.

We investigate the relationship between the time decay of the solutions u of the Navier–Stokes system on a bounded open subset of $\mathbb{R}^{N}$ and the time decay of the right-hand sides f. In suitable function spaces, we prove that u always inherits at least part of the decay of f, up to exponential, and that the decay properties of u depend only upon the amount and type (e.g., exponential, or power-like) of decay of f. This is done by first making clear what is meant by “type” and “amount” of decay and by next elaborating upon recent abstract results pointing to the fact that, in linear and nonlinear PDEs, the decay of the solutions is often intimately related to the Fredholmness of the differential operator.