2.5 Free boundary problems2 Local existence2.3 Questions of differentiability

2.4 Matter fields

  Analogues of the results for the vacuum Einstein equations given in section 2.2 are known for the Einstein equations coupled to many types of matter model. These include perfect fluids, elasticity theory, kinetic theory, scalar fields, Maxwell fields, Yang-Mills fields and combinations of these. An important restriction is that the general results for perfect fluids and elasticity apply only to situations where the energy density is uniformly bounded away from zero on the region of interest. In particular they do not apply to cases representing material bodies surrounded by vacuum. In cases where the energy density, while everywhere positive, tends to zero at infinity, a local solution is known to exist, but it is not clear whether a local existence theorem can be obtained which is uniform in time. In cases where the fluid has a sharp boundary, ignoring the boundary leads to solutions of the Einstein-Euler equations with low differentiability (cf. section  2.3), while taking it into account explicitly leads to a free boundary problem. This will be discussed in more detail in section  2.5 . In the case of kinetic or field theoretic matter models it makes no difference whether the energy density vanishes somewhere or not.

2.5 Free boundary problems2 Local existence2.3 Questions of differentiability

image Local and Global Existence Theorems for the Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr-2000-1
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