Shock waves, part 3

In a previous post I wrote about shock waves in fluids, including the case that they are described by the Einstein-Euler equations for a self-gravitating fluid in general relativity. I mentioned there a result of Fredrik Ståhl and myself proving that smooth solutions of the Einstein-Euler system can lose regularity in the course of their time evolution. This was done in the framework of spacetimes with plane symmetry. Here I want to describe some complementary results which were recently obtained by Philippe LeFloch and myself. These new results concern the existence of global weak solutions in situations where shocks may be present. This work is done under the assumption of Gowdy symmetry, which is weaker than plane symmetry. It allows the presence of gravitational waves, which plane symmetry does not. It uses time coordinates different from the constant mean curvature (CMC) coordinate used in the work with Ståhl. This difference in the time coordinates makes it difficult to relate the results of the two papers directly. It would be interesting to adapt the results of either of these papers to the time coordinates of the other.

In the paper with LeFloch we use coordinates (areal, conformal) which have previously been used in analysing analogous problems for vacuum spacetimes or spacetimes where the matter content is described by collisionless kinetic theory. A big difference is the weak regularity. One effect of this is that while in the given context it has been possible to prove global existence theorems for the initial value problem, nothing is known about the uniqueness of the solutions in terms of initial data. It should, however, be noted that in the corresponding analytical framework uniqueness is not even known for a one-dimensional non-relativistic fluid without gravity. Another new element introduced by the use of weak solutions is that it is only possible to evolve in one time direction. This model is not reversible, a fact implemented mathematically by the imposition of entropy inequalities. One of the results obtained concerns a forever expanding cosmological model. The other one concerns a contracting model which ends in a singularity. The second is not a global existence result in the conventional sense but it can be thought of as saying that the solution can be extended until certain specific things happen (a big crunch singularity).

To finish this post I want to indicate the type of regularity of the solutions obtained. I only state this roughly – more precise information can be found in the paper. The energy density and momentum density of the fluid is integrable in space, with the norms locally bounded in time. The quantities parametrizing the spacetime metric have first order derivatives which are square integrable in space. These conditions allow for jump discontinuities in the energy density which is what comes up in shock waves. It also allows singularities of Dirac type in the metric, corresponding to what are often called impulsive gravitational waves.

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