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#### The Stability of the Minkowski space for the Einstein-Vlasov system

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##### Citation

Fajman, D., Joudioux, J., & Smulevici, J. (2021). The Stability of the Minkowski
space for the Einstein-Vlasov system.* Analysis & PDE,* *14*(2),
425-531. doi:10.2140/apde.2021.14.425.

Cite as: https://hdl.handle.net/21.11116/0000-0005-1BEC-4

##### Abstract

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 developed in [FJS15;

FJS17]. 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 non-linear 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.

solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use

the vector field and modified vector field techniques developed in [FJS15;

FJS17]. 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 non-linear 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.