Leila Azem, Ridha Selmi
abstract:
A regularized periodic three-dimensional Boussinesq system is studied. The
existence, uniqueness and continuous dependance with respect to the initial data
of weak and strong solutions are proved under the minimum regularity
requirements. The main novelty is that these solutions are global in time. Also,
convergence results of the unique weak solution and the unique strong solution
of the three-dimensional regularized Boussinesq system to solutions of the
three-dimensional Boussinesq system are established as the regularizing
parameter $\alpha$ vanishes. We overcome the main difficulty caused by the
singular dependance of the energy estimates on the regularizing parameter; as if
it vanishes, the energy estimates blow up. The proofs use energy methods and
compactness arguments.
Mathematics Subject Classification: Primary: 35A01, 35A02, 35B40, 35B25, 35B30. Secondary: 35B10, 35B45
Key words and phrases: Three-dimensional Boussinesq system, regularization, existence of global in time solution, uniqueness, continuous dependence on initial data, asymptotic behavior, singular perturbation, weak solution, strong solution