We analyze the two-field formulation of the quasi-static Biot's equations in bounded domains 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：本文通过下确界-上确界理论分析了有界域中准静态Biot方程的双场表述形式。为此，我们利用该方程组的等效四场表述，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，这意味着解所具备的正则性与数据要求的正则性相匹配，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的相应额外正则性。这些结果对于设计具有精确稳定性和误差估计的离散化方案及其分析具有重要价值。