We prove the global stability of the Minkowski space viewed as the trivial
solution of the Einstein–Vlasov system. To estimate the Vlasov field, we
use the vector field and modified vector field techniques we previously developed
in 2017. In particular, the initial support in the velocity variable does not
need to be compact. To control the effect of the large velocities, we identify
and exploit several structural properties of the Vlasov equation to prove
that the worst nonlinear terms in the Vlasov equation either enjoy a form
of the null condition or can be controlled using the wave coordinate gauge.
The basic propagation estimates for the Vlasov field are then obtained
using only weak interior decay for the metric components. Since some of
the error terms are not time-integrable, several hierarchies in the commuted
equations are exploited to close the top-order estimates. For the Einstein
equations, we use wave coordinates and the main new difficulty arises from
the commutation of the energy-momentum tensor, which needs to be rewritten
using the modified vector fields.
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