论文标题
经典的BousSinesQ系统重新审视
The classical Boussinesq system revisited
论文作者
论文摘要
在这项工作中，我们对M. E. Schonbek [11]进行了研究，该研究涉及经典BoussinesQ系统的全球熵弱解决方案以及C. J. Amick [1]对这些解决方案的规律性的研究。我们建议由“分形”操作员正规化（即，由类型$ $ε|ξ|^λ，\，（ε，λ）\ in \，\ mathbb {r} _+\ \ \ \ times] 0,2] 0,2] $定义的差异操作员。我们首先表明，正规化系统在全球范围内无条件地在$ h^s（\ mathbb {r}）的Sobolev空间中，\，\，s> \ frac {1} {2} {2}，$，在正则化参数$（ε，λ）\ in \ in \ in \ in \ in \，\ mathbbbbbbbbbb in cliforceply中。结果，我们在这种规律性级别上获得了经典的BousSinesQ系统的全球体系良好，以及正规化系统解决方案解决方案的强烈拓扑的融合，因为参数E到0。 $ζ_0$在Orlicz类中作为常规解决方案的较弱限制。
In this work, we revisit the study by M. E. Schonbek [11] concerning the problem of existence of global entropic weak solutions for the classical Boussinesq system, as well as the study of the regularity of these solutions by C. J. Amick [1]. We propose to regularize by a "fractal" operator (i.e. a differential operator defined by a Fourier multiplier of type $ε|ξ|^λ, \, (ε,λ) \in\,\mathbb{R}_+\times ] 0,2]$). We first show that the regularized system is globally unconditionally well-posed in Sobolev spaces of type $H^s(\mathbb{R}),\,s > \frac {1}{2},$, uniformly in the regularizing parameters $(ε,λ) \in\,\mathbb{R}_+\times ]0,2]$. As a consequence we obtain the global well-posedness of the classical 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 classical Boussinesq equation as the parameter e goes to 0. In a second time, we prove the existence of low regularity entropic solutions of the Boussinesq equations emanating from $u_0 \in H^1$ and $ζ_0$ in an Orlicz class as weak limits of regular solutions.
