Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2006-07-18
Nonlinear Sciences
Pattern Formation and Solitons
27 Pages, 7 figures are not present in this version. See http://www.columbia.edu/~grs2103/ for a PDF with figures. Submitted t
Scientific paper
10.1088/0951-7715/20/1/003
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.
Simpson Gideon
Spiegelman Marc
Weinstein Michael I.
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