溶胀多孔弹性固体中稳态Darcy/Fick流动的变分方法
Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids
摘要：
结合静态情况下的变分方法和Schauder的不动点论证，证明了在小应变下，由于Fick或Darcy定律，溶剂或流体扩散的弹性介质中稳态的存在性.在平面变形中，问题在于与连续性方程耦合的力平衡，并且基本算子是非势和非伪单调的，使得常规方法不适用。在先进的变体，带电的多组分流通过带电的弹性固体处理，采用鞍点型的临界点。最后，还讨论了涉及传热方程的非等温变式。
英文摘要：
Existence of steady states in elastic media at small strains with diffusion of a solvent or fluid due to Fick's or Darcy's laws is proved by combining usage of variational methods inspired from static situations with Schauder's fixed-point arguments. In the plain variant, the problem consists in the force equilibrium coupled with the continuity equation, and the underlying operator is non-potential and non-pseudomonotone so that conventional methods are not applicable. In advanced variants, electrically-charged multi-component flows through an electrically charged elastic solid are treated, employing critical points of the saddle-point type. Eventually, anisothermal variants involving heat-transfer equation are treated, too.
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