The Cauchy Problem and Decay Rates for Strong Solutions of a Boussinesq System

where the unknown are u, θ, π which denote, respectively, the velocity field, the scalar temperature and the scalar pressure. Data are the positive constants ν, χ, respectively, the viscosity and the thermal conductivity coefficients and the function f the external force field, and a(x), b(x), respectively, represent the initial velocity and initial temperature. The main objective of this work is to obtain a decay rate of derivatives for the strong solutions to the Cauchy problem (1.1). For this, we will consider the usual Lebesgue spaces Lp(R3) with the usual norms | · |p. We will denote L σ (R3) the closure ofC∞ 0,σ(R)= {v ∈ C∞ 0 ;divv = 0} in Lp(R3). We will denote too by LP(0,T ;Lq(R3)) the Banach space, classes of functions defined a.e. in [0,T] on Lq(R3), that are Lp-integrable in the sense of Bochner. For more details see [1, 3]. We observe that this model of fluids includes as a particular case the classical NavierStokes equations, which has been thoroughly studied (see, e.g., [7, 8]). Rojas Medar and