Abstract.In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as t → ∞ in the sense that the energy and the L p -norms of the velocity field grow to infinity for large time for 1 ≤ p < 3. In the case of strong solutions we provide sharp estimates, both from above and from below, and explicit asymptotic profiles. We also show that solutions arising from (u 0 , θ 0 ) with zero-mean for the initial temperature θ 0 have a special behavior as |x| or t tends to infinity: contrary to the generic case, their energy dissipates to zero for large time.

We show that the solutions to the non-stationary Navier-Stokes equations in R d (d = 2, 3) which are left invariant under the action of discrete subgroups of the orthogonal group O(d) decay much faster as |x| → ∞ or t → ∞ than in generic case and we compute, for each subgroup, the precise decay rates in space-time of the velocity field.