Ridha Selmi, Mounia Zaabi
abstract:
We prove existence of weak solution to a regularized Boussinesq system in
Sobolev spaces under the minimal regularity to the initial data. Continuous
dependence on initial data (and then uniqueness) is proved provided that the
initial fluid velocity is mean free. If the temperature is also mean free, we
prove that the solution decays exponentially fast, as time goes to infinity.
Moreover, we show that the unique solution converges to a Leray-Hopf solution of
the three-dimensional Boussinesq system, as the regularizing parameter alpha
vanishes. The mean free technical condition appears because the nonlinear part
of the fluid equation is subject to regularization. The main tools are the
energy methods, the compactness method, the Poincar\'e inequality and some
Gr\"onwall type inequalities. To handle the long time behaviour, a time
dependent change of function is used.
Mathematics Subject Classification: Primary 35A05, 35B30, 35B40; Secondary 35B10, 35B45
Key words and phrases: Three-dimensional periodic Boussinesq system, weak solution, regularization, existence, uniqueness, convergence, asymptotic behavior, long time behavior, mean free