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For this and other reasons discussed below, we consider a weak formulation of the problem. As we have noted, the effect of the mapping is seen in additional coefficients appearing in the equations. The values of these coefficients must be interpreted through the inverse map T~l(q) to determine the geometric parameters. Another difficulty arises from the oscillatory nature of the time domain data. Varying some parameters in the model has the effect of changing the time at which reflected signals are detected by varying the distance the propagated waves travel (to and from the unknown boundary F(g)) and their speed of propagation.

For this and other reasons discussed below, we consider a weak formulation of the problem. As we have noted, the effect of the mapping is seen in additional coefficients appearing in the equations. The values of these coefficients must be interpreted through the inverse map T~l(q) to determine the geometric parameters. Another difficulty arises from the oscillatory nature of the time domain data. Varying some parameters in the model has the effect of changing the time at which reflected signals are detected by varying the distance the propagated waves travel (to and from the unknown boundary F(g)) and their speed of propagation.

We further assume that a can be written a(t, z) = In(z)a(t), Well-Posedness 39 where a € Hl(Q,T) with d(t) > 0, a(t] < 0, so that a is constant in z on£l and positive nonincreasing in t. Then the operator A is the infinitesimal generator of a Co-semigroup on Z\ and hence on the equivalent space Z. To prove this theorem, we use the Lumer-Phillips theorem [Paz83, p. 14]. Since Z\ is a Hilbert space, it suffices to argue that for some AQ, A — \Q! is dissipative in Z\ and H(\I — A] = Z\ for some A > 0, where 7£(AJ — A} is the range of XI — A.

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an introduction to magnetohydrodynamcs by Roberts


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