ABSTRACT
We are interested in solving the three-dimensional incompressible Navier-Stokes system in the whole space, say
It is well-known that local (or global under smallness assumptions on u 0) well-posedness holds in various functional spaces ([19, 18, 3, 22]). A classical example is provided by the space of continuous in time solutions taking values in the Sobolev space say
Moreover, we have at our disposal a persistence result, namely: if the initial data u0 is not only small in but also belongs to H 1, then the above mentioned solution u is globally continuous in time with values in H 1 . To prove such a result it is enough to show that the H 1norm of the solution is a Lyapunov function, which means a decreasing in time one. More accurately, in the celebrated paper by T.Kato and H.Fujita [19] the following inequality is proven :
that immediately provides the announced decreasing in time of the homogeneous norm
as long as is small enough. On the other hand the L 2 norm of the solution u decreases as well, which allows to conclude. T.Kato [20] extended this kind of result, actually proving the following inequality :
(0.1)
(0.2)
where and Q p is a quantity depending on u and its derivatives. In other words, a new family of Lyapunov functions is derived, and they do not necessarily arise from an energy norm.
