  In this paper, we study the dynamics of fluids in porous media governed by
Darcy's law: the Muskat problem. We consider the setting of two immiscible
fluids of different densities and viscosities under the influence of gravity in
which one fluid is completely surrounded by the other. This setting is gravity
unstable because along a portion of the interface, the denser fluid must be
above the other. Surprisingly, even without capillarity, the circle-shaped
bubble is a steady state solution moving with vertical constant velocity
determined by the density jump between the fluids. Taking advantage of our
discovery of this steady state, we are able to prove global in time existence
and uniqueness of dynamic bubbles of nearly circular shapes under the influence
of surface tension. We prove this global existence result for low regularity
initial data. Moreover, we prove that these solutions are instantly analytic
and decay exponentially fast in time to the circle.
