In this paper we introduce and analyze, up to our knowledge for the first time, Banach spaces-based mixed variational formulations for nearly incompressible linear elasticity and Stokes models. Our interest in this subject is motivated by the respective need that arises from the solvability studies of nonlinear coupled problems in continuum mechanics that involve these equations. We consider pseudostress-based approaches in both cases and apply a suitable integration by parts formula for ad-hoc Sobolev spaces to derive the corresponding continuous schemes. We utilize known and new preliminary results, along with the Babuška-Brezzi theory in Banach spaces, to establish the well-posedness of the formulations for a particular range of the indexes of the Lebesgue spaces involved. Among the aforementioned new results from us, we highlight the construction of a particular operator mapping a tensor Lebesgue space into itself, and the generalization of a classical estimate in ${\mathrm{L}}^2$ for deviatoric tensors, which plays a key role in the Hilbertian analysis of linear elasticity, to arbitrary Lebesgue spaces. No discrete analysis is performed in this work.

在本文中，我们首次介绍和分析了基于 Banach 空间的几乎不可压缩线性弹性和 Stokes 模型的混合变分公式。我们对这个主题的兴趣源于对涉及这些方程的连续介质力学中非线性耦合问题的可解性研究的各自需求。我们在这两种情况下都考虑基于伪应力的方法，并为临时 Sobolev 空间应用适当的分部积分公式来推导出相应的连续方案。我们利用已知的和新的初步结果，以及 Banach 空间中的 Babuška-Brezzi 理论，为所涉及的 Lebesgue 空间的特定索引范围建立公式的适定性。在上述我们的新成果中，${\mathrm{升}}^2$对于偏张量，它在线性弹性的希尔伯特分析中起关键作用，到任意勒贝格空间。在这项工作中没有进行离散分析。