Large time decay of solutions to the Boussinesq system with fractional dissipation

In this paper we extend the results of Brandolese and Schonbek [2] to the Boussinesq system with fractional dissipation. Let Λα and Λβ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We show that if the initial data (u0,θ0)∈Lσ2×(L1∩L2), then ‖θ(t)‖≤C(1+t)−d2β, and ‖u(t)‖≤C(1+t)max⁡{0,1−d2β} if β≠d2, ‖u(t)‖≤Cln(2+t) if β=d2; if we additionally assume ∫θ0=0 and θ0∈L11, then ‖θ(t)‖≤C(1+t)−d+22β and ‖u(t)‖→0 as t→∞. Comparing [2], we don't need to assume that ‖θ0‖1 is sufficiently small when β∈(0,d+12).