Global Solutions to Bubble Growth in Porous Media

TL;DR

This paper analyzes a free boundary problem modeling fluid injection in porous media, providing solutions using generalized Newtonian potentials and characterizing long-term bubble growth based on initial conditions.

Contribution

It introduces a novel approach to solving the moving boundary problem using generalized Newtonian potentials, linking initial potential properties to long-term behavior.

Findings

Bubble occupies entire space as time tends to infinity under specific initial conditions.

Solutions are expressed in terms of the time-derivative of generalized Newtonian potentials.

Long-term growth depends on the initial bubble's potential being quadratic.

Abstract

We study a moving boundary problem modeling an injected fluid into another viscous fluid. The viscous fluid is withdrawn at infinity and governed by Darcy's law. We present solutions to the free boundary problem in terms of time-derivative of a generalized Newtonian potentials of the characteristic function of the bubble. This enables us to show that the bubble occupies the entire space as the time tends to infinity if and only if the internal generalized Newtonian potential of the initial bubble is a quadratic polynomial.