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 The viscous fluid is withdrawn at infinity and governed by Darcy's law.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.
Remarks on the global solutions of 3-D Navier-Stokes system with one slow variable
Structure of Helicity and Global Solutions of Incompressible Navier-Stokes Equation
On global solutions and blow-up for Kuramoto-Sivashinsky-type models, and well-posed Burnett equations
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