UC Berkeley

We study nonlinear partial differential equations describing Hele-Shaw flows for which the source of the fluid, which classically is at a single point, instead expands as the moving boundary uncovers new sources. This problem is reformulated as a type of quasi- variational inequality, for which we show there exists a unique weak solution. Our procedure utilizes a Baiocchi integral transform. However, unlike in traditional applications, the inequality solved by the transformed variable continues to depend on the free boundary, and thus standard theory of variational inequalities cannot be immediately applied. Instead, we develop convergent time-stepping scheme. As a further application, we generalize and study a related problem with a growing density function obeying a hard constraint on the maximum density.