Abstract
We consider the Muskat Problem with surface tension in two dimensions over the real line, with H s initial data and allowing the two fluids to have different constant densities and viscosities. We take the angle between the interface and the horizontal, and derive an evolution equation for it. Via energy methods, it has been shown that a unique solution \({\theta}\) exists locally and can be continued while \({||\theta||_{s}}\) remains bounded and the arc chord condition holds. We prove that when both fluids have the same viscosity and the initial data is sufficiently small, the energy estimate is dominated by second-order dissipative terms. As a result, the energy is non-increasing, and that the resulting solution \({\theta}\) exists globally in time.
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Communicated by G.P. Galdi
This research was partially supported by NSF Grant DMS-1500916.
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Tofts, S. On the Existence of Solutions to the Muskat Problem with Surface Tension. J. Math. Fluid Mech. 19, 581–611 (2017). https://doi.org/10.1007/s00021-016-0297-y
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DOI: https://doi.org/10.1007/s00021-016-0297-y