The Boussinesq system revisited

In this work, we revisit the study by Schonbek (1981 J. Differ. Equ. 42 325–52) concerning the problem of existence of global entropic weak solutions for a specific Boussinesq system, as well as the study of the regularity of these solutions by Amick (1984 J. Differ. Equ. 54 231–47). We propose to study a regularized variant of this Boussinesq system, obtained by adding a 'fractal' operator (i.e. a differential operator defined by a Fourier multiplier of type ${\epsilon}\vert \xi {\vert }^{\lambda },\left({\epsilon},\lambda \right)\in {\mathbb{R}}_{+}{\times}$] 0, 2]) to the equation of the height of the water column. We first show that the regularized system is globally unconditionally well-posed in Sobolev spaces of type ${H}^{s}\left(\mathbb{R}\right),s{ >}\frac{1}{2},$ uniformly in the regularizing parameters $\left({\epsilon},\lambda \right)\in {\mathbb{R}}_{+}{\times}$]0, 2]. As a consequence we obtain the global unconditional well-posedness of this Boussinesq system at this level of regularity as well as the convergence in the strong topology of the solution of the regularized system towards the solution of the Boussinesq system as the parameter goes to 0. Finally, we prove the existence of low regularity entropic solutions of the Boussinesq equations emanating from u₀ ∈ H¹ and ζ₀ in an Orlicz class as weak limits of regular solutions.