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Degenerate dispersive equations arising in the study of magma dynamics
arXiv.org, v 20(1), pp 21-49
18 Jul 2006
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Abstract
An outstanding problem in Earth science is understanding the method of transport of magma in the Earth's mantle. Models for this process, transport in a viscously deformable porous media, give rise to scalar degenerate, dispersive, nonlinear wave equations. We establish a general local well-posedness for a physical class of data (roughly \(H^1\)) via fixed point methods. The strategy requires positive lower bounds on the solution. This is extended to global existence for a subset of possible nonlinearities by making use of certain conservation laws associated with the equations. Furthermore, we construct a Lyapunov energy functional, which is locally convex about the uniform state, and prove (global in time) nonlinear dynamic stability of the uniform state for any choice of nonlinearity. We compare the dynamics to that of other problems and discuss open questions concerning a larger range of nonlinearities, for which we conjecture global existence.
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Details
- Title
- Degenerate dispersive equations arising in the study of magma dynamics
- Creators
- Gideon Simpson - Columbia UniversityMarc Spiegelman - Columbia UniversityMichael Weinstein - Columbia University
- Publication Details
- arXiv.org, v 20(1), pp 21-49
- Publisher
- Cornell University Library, arXiv.org; Ithaca
- Resource Type
- Other
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000243750500003
- Scopus ID
- 2-s2.0-33947625062
- Other Identifier
- 991019297086904721
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- Web of Science research areas
- Mathematics, Applied
- Physics, Mathematical