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Weak solutions for a poro-elastic plate system
Applicable Analysis ( IF 1.1 ) Pub Date : 2021-07-21 , DOI: 10.1080/00036811.2021.1953483
Elena Gurvich 1 , Justin T. Webster 1
Affiliation  

ABSTRACT

We consider a recent plate model obtained as a scaled limit of the three-dimensional Biot system of poro-elasticity. The result is a ‘2.5’-dimensional linear system that couples traditional Euler–Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We allow the permeability function to be time dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so-called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions for implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability function. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate and exposits a variety of interesting related models and associated analytical investigations.



中文翻译:

多孔弹性板系统的弱解决方案

摘要

我们考虑最近获得的板模型作为孔隙弹性的三维 Biot 系统的比例极限。结果是一个“2.5”维线性系统,它将传统的欧拉-伯努利板动力学耦合到三维压力方程,其中扩散仅横向作用。我们允许渗透函数是时间相关的,使问题成为非自治的,并取消了许多标准抽象理论的资格。在所谓的准静态情况下定义弱解,并将问题抽象地构建为隐含的退化演化问题。利用隐式演化方程的弱解理论,我们得到解的存在性。唯一性是在对渗透函数规律性的附加假设下获得的。我们在附录中讨论惯性情况,通过半群论。这里的工作为多孔弹性板提供了弱解的基线理论,并展示了各种有趣的相关模型和相关的分析研究。

更新日期:2021-07-21
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