Global Well-Posedness and Convergence Results for the 3D-Regularized Boussinesq System

Abstract

Abstract. Analytical study to the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solution with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray-Hopf solution of the Boussinesq system are established as the regularizing parameter α vanishes. The proofs are done in the frequency space and use energy methods, Arselà-Ascoli compactness theorem and a Friedrichs like approximation scheme.

Well‐posedness and convergence results for strong solution to a 3D‐regularized Boussinesq system

“…In [1], we proved that weak solution is unique. As strong solutions are also weak, one deduces uniqueness for the formers.…”

“…Using the energy estimate for weak solution [1] and the expression of the function ˛g iven by equation (1.7), we infer that…”

“…To study the existence and the regularity of the strong solution, we first approximate .Bq˛/ by system .Bq˛/ n , for which we can apply the classical theory of ordinary differential equations in order to construct approximate solutions, as in [1], for example. Next, we obtain uniform estimates, with respect to the approximating parameter n, on the approximate solutions.…”

“…The data  0 and u 0 are initial temperature and free divergence velocity. In [1], motivations behind considering .Bq˛/ are explained, and a wide review of related lecture is given. As for physical motivation, the Boussinesq system is used as a toy model for geophysical fluids whenever rotation and stratification play important roles.…”

“…0. This is one of the differences with the convergence problem for the weak solution to the three-dimensional-regularized Boussinesq system, in [1], where dependence on˛was a polynomial type. Moreover, to prove the convergence results, one needs to derive uniform estimates on˛; this is not an easy task because of the singular terms such as 1 4 .…”