Local Strong Solution Of A Three Dimensional Boussinesq System With Dissipation Energy And A Blow Up Criterion

We consider a system of Boussinesq equations with dissipation energy on athree dimensional smooth bounded domain. The system consists of the nonhomo-geneous incompressible Navier-Stokes equations and an internal energy equation,which constains a strong nonlinear term|D(u)|~2, where D(u) is the deformationtensor. We prove that the initial-boundary-value problem has a local strong solu-tion. We also prove a blow-up criterion, namely, if a strong solution blow-up at afnite time T, then the L~2(0, T; L~∞) norm of the deformation tensor is infnite.For proving the existence of a strong solution, we use the classical Galerkinmethod to solve the linearized equations. Then we iterate the approximate solu-tions to get a solution of the original nonlinear problem. For proving the blow-upcriterion, we adopt the approach used by Huang-Li-Xin in proving the Beale-Kato-Majda blow-up criterion for the compressible Navier-Stokes equations.