Well-posedness of two-phase Darcy flow in 3D

Ambrose, David

doi:10.1090/S0033-569X-07-01055-3

Author: David M. Ambrose
Journal: Quart. Appl. Math. 65 (2007), 189-203
MSC (2000): Primary 35Q35
DOI: https://doi.org/10.1090/S0033-569X-07-01055-3
Published electronically: February 12, 2007
MathSciNet review: 2313156
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Abstract: We prove the well-posedness, locally in time, of the motion of two fluids flowing according to Darcy’s law, separated by a sharp interface in the absence of surface tension. We first reformulate the problem using favorable variables and coordinates. This results in a quasilinear parabolic system. Energy estimates are performed, and these estimates imply that the motion is well-posed for a short time with data in a Sobolev space, as long as a condition is satisfied. This condition essentially says that the more viscous fluid must displace the less viscous fluid. It should be true that small solutions exist for all time; however, this question is not addressed in the present work.

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