Interface evolution: the Hele-Shaw and Muskat problems

• 成果类型: Article
• 署名作者: Cordoba, Antonio; Cordoba, Diego; Gancedo, Francisco
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.1.10
• 发表日期: 2011
• 页码: 477-542
• 关键词: maximum principle; porous-medium; FLOW; posedness; EXISTENCE; fluids

摘要

We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy's law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two-dimensional Muskat problem or the two-phase He le-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.