Abstract We study the zero surface tension limit of three-dimensional interfacial Darcy flow. We start with a proof of well-posedness of three-dimensional interfacial Darcy flow for any positive value of the surface tension coefficient. The primary tool for this well-posedness proof is an energy estimate. The time of existence for these solutions will, in general, go to zero with the surface tension parameter. However, in the case that a stability condition is satisfied by the initial data, we prove an additional energy estimate, establishing that the time of existence can be made uniform in the surface tension parameter. Then, an additional estimate allows the limit to be taken as surface tension vanishes, demonstrating that three-dimensional interfacial Darcy flow without surface tension is the limit of three-dimensional interfacial Darcy flow with surface tension as surface tension vanishes. This provides a new proof of existence of solutions for the problem without surface tension.

中文翻译：

---

三维界面达西流的零表面张力极限

摘要 我们研究了三维界面达西流的零表面张力极限。我们首先证明对于任何表面张力系数的正值，三维界面达西流的适定性证明。这种适定性证明的主要工具是能量估计。这些解决方案的存在时间通常会随着表面张力参数变为零。然而，在初始数据满足稳定条件的情况下，我们证明了额外的能量估计，确定存在时间可以在表面张力参数中保持一致。然后，额外的估计允许在表面张力消失时采用极限，证明没有表面张力的三维界面达西流是当表面张力消失时具有表面张力的三维界面达西流的极限。这为没有表面张力的问题的解决方案的存在提供了新的证明。