Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations

In this paper we deal with the asymptotic behavior, in the space-time variables, of weak and strong solutions to the Navier-Stokes system. For an incompressible viscous fluid which fills the whole space R n , in the absence of external forces, the @tu + r · (u u) = u r p u(x,0) = a(x) div(u) = 0. Here u : R n ◊ (0,1(! R n (n 2) denotes the velocity field and p(x,t) is the pressure. Starting with the pioneering work of Leray (21), a considerable number of papers is concerned with questions related to the large-time behavior of the L 2 -norm of u(t). The problem of finding optimal decay rates for the energy of generic weak solutions is now well understood. Indeed, Wiegner (41) showed that ||u(t)||2 C(1 +t) (0 < (n + 2)/4), if such decay holds for the solution e t a of the heat equation starting with the same data. This improved previous results by Kato (17), Schonbek (29) and Kaijkiya- Miyakawa (19). The bound on is now known to be optimal: optimality was first discussed in (30) and, more recently, in (28), (13), (14) with dierent methods. However, exceptional flows which decay much faster do exist. For example, it is known since a long time that, in dimension n = 2, there exists a very particular and explicit solution of the Navier-Stokes equations with radial vorticity. This condition on the vorticity implies that the nonlinearity has the potential form (i.e. r ·(u u) = r p), so that u is also a solution of the homogeneous heat equation. It was pointed out by Majda and Schonbek that for such flow ||u(t)||2 has an exponential decay at infinity (see e.g. (30), (10), (28)). In dimension 2, no other examples with such a property seem to be known. Similar flows with exponential decay exist in higher even dimension and a general method for their construction is described in (32). All these solutions, sometimes called generalized Beltrami flows, turn out to solve simultaneously (NS) and the heat equation. As discussed in (32), it seems impossible to adapt these examples to the n = 3 case or for general odd dimensions.