We analyze the two-field formulation of the quasi-static Biot's equations by means of the inf-sup theory. For this purpose, we exploit an equivalent four-field formulation of the equations, introducing the so-called total pressure and total fluid content as independent variables. We establish existence, uniqueness and stability of the solution. Our stability estimate is two-sided and robust, meaning that the regularity established for the solution matches the regularity requirements for the data and the involved constants are independent of all material parameters. We prove also that additional regularity in space of the data implies, in some cases, corresponding additional regularity in space of the solution. These results are instrumental to the design and the analysis of discretizations enjoying accurate stability and error estimates.

翻译：我们通过inf-sup理论分析了准静态Biot方程的两场表述形式。为此，我们利用该方程的一个等效四场表述形式，引入所谓总压力和总流体含量作为独立变量。我们建立了解的存在性、唯一性和稳定性。我们的稳定性估计是双向且鲁棒的，这意味着为解建立的正则性与数据的正则性要求相匹配，且所涉及的常数与所有材料参数无关。我们还证明了在某些情况下，数据在空间上的额外正则性会引致解在空间上相应的额外正则性。这些结果对于设计和分析具有精确稳定性及误差估计的离散化方案至关重要。