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 ] 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 uniformly in the regularizing parameters ]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 ϵ goes to 0. Finally, we prove the existence of low regularity entropic solutions of the Boussinesq equations emanating from u ₀ ∈ H ¹ and ζ ₀ in an Orlicz class as weak limits of regular solutions.

在这项工作中，我们重新审视了 Schonbek (1981 J. Differ. Equ . 42 325–52) 关于特定 Boussinesq 系统全局熵弱解存在问题的研究，以及对这些解的规律性的研究作者：Amick (1984 J. Differ. Equ . 54 231–47)。我们建议研究这个 Boussinesq 系统的正则化变体，通过将“分形”算子（即由类型为] 0, 2]的傅立叶乘法器定义的微分算子）添加到水柱高度方程中而获得。我们首先表明，正则化系统在正则化参数中均匀类型的 Sobolev 空间中全局无条件适定]0, 2]。因此，我们获得了这个 Boussinesq 系统在这个规则水平上的全局无条件适定性，以及当参数ϵ变为 0 时，正则化系统的解向 Boussinesq 系统的解的强拓扑的收敛性. 最后，我们证明了在 Orlicz 类中从u ₀ ∈ H ¹和ζ ₀发出的 Boussinesq 方程的低正则熵解是正则解的弱极限。