where is the velocity vector field and is the pressure.

Dealing with the Cauchy problem with initial data
,
it is possible to rewrite the system
in the following integral expression:

where is the heat semigroup defined by the convolution with the Gauss kernel:

and is the projection operator on the divergence-free vector fields, defined by the matrix:

where are the Riesz transforms for .

In fact the integral and the differential systems are equivalent in this framework, as it is shown in [6] under slight assumptions verified by a large class of spaces.

Given a Banach space , the problem of the existence and the uniqueness of a function solution of (1) in (a so-called mild solution) is normally approached by a fixed point argument. It is now well known ([6], [11], [9]) that for a given the only solution in belongs to every space with for all its existence time. We are now interested in investigating the following problem: if the data have some extra properties, does the solution keep verifying them at least on a small interval of time?

In [6] it was already established that if , the Hardy space, then the solution satisfies for . We will consider here a particular case of the previous one where the Laplacian of the data is a ``molecule'' for the Hardy space, and so well-localized in the space variables; we will prove then that the solution verifies the same property, namely its diffusion term keeps on being well-localized at the beginning of the motion.

In the following the reference space will always be .

3=2 3 by 1 3
More precisely,
let
,
and let us introduce the following functional
space:

The space normed by is a Banach space.

The theorem we are going to prove is the following one.

3=2 3 by 1 3

**Theorem .**

*Let
and
Then the mild solution of the Navier-Stokes equations
is also in
for .*

We would like to stress that the solution we obtain decays pointwise in space like for every (see also proposition 2). In the particular case , this decay property may also be recovered by the results of Miyakawa in [12], who establishes a pointwise space-time asymptotic behavior with respect to space-time variables and for a particular class of data.

For other results close to these topics see also [1], [3], [4], [7], [13], [15], [17].