Overview
This research addresses a Hele-Shaw type free boundary problem, examining its characteristics in general spatial dimensions. The investigation centers on establishing foundational mathematical properties for this problem, specifically focusing on its comparison principle and well-posedness.
Research Context
Hele-Shaw problems are a class of free boundary problems that describe the motion of a fluid in a narrow gap between two parallel plates. The present study extends the analysis of such problems to general spatial dimensions. A key aspect of this work involves scenarios where the velocity of the free boundary can be negative, which is a condition accommodated by the research hypotheses.
Approach
The method employed for this investigation involves the application of a notion of viscosity flows. This approach facilitates the analysis of the free boundary problem and is crucial for establishing the stated mathematical properties. The hypotheses formulated for this work are designed to allow for the free boundary velocity to assume negative values.
Findings
- A comparison principle was established for the Hele-Shaw type free boundary problem in general spatial dimensions.
- Well-posedness for this problem was established using the concept of viscosity flows.
- The hypotheses developed for this study permit negative values for the free boundary velocity.