We study the flow of magma, partially molten rock within the earth's interior.
1) Beginning with a microscopic model of two interpenetrating viscous and
incompressible fluids (flow in a deformable porous media) we derive, via homogenization methods,
PDEs for a macroscopic compressible solid (matrix) and incompressible fluid (magma).
These models are consistent with the class of PDEs first introduced by D. McKenzie.
The advantage of our approach is that
we have removed closure assumptions and characterize effective macroscopic parameters:
e.g. permeability and viscosity, in terms of computationally tractable ``cell problems'',
defined by the microscopic model.
2) Bulk parameters are studied for different microscopic geometries. In particular, new results are obtained
for the bulk viscosity in terms of the porosity.
3) In the low porosity limit, realistic in some regimes,
these PDEs reduce to nonlinear, degenerate and dispersive
wave equations, having an emergent length scale (compaction length).
We prove that these evolution equations have
coherent asymptotically stable solitary wave structures.
This is joint work with Gideon Simpson and Marc Spiegelman.
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