Birth of a Theorem: A Mathematical Adventure (41 page)

Assume that W and f
⁰
satisfy the generalized Penrose linear stability condition: for all
if we assume further that
and for all
then for all
such that
we have

Now assume an initial position and velocity profile, f
_i
(
x
,
v
)
≥
0
, very close to the analytic state f
⁰
, in the sense that its Fourier transform
with respect to position and velocity satisfies

with
λ,
μ
>
0
, and
ε
>
0
small enough.

Then there exist analytic profiles f
_+∞
(
v
), f
_−∞
(
v
)
such that the solution of the nonlinear Vlasov equation, with interaction potential W and initial datum f
_i
at time t
=
0
, satisfies

weakly; more precisely, in the sense of a pointwise, exponentially fast convergence of Fourier modes.

The convergence rate of the nonlinear equation is arbitrarily close to the convergence rate of the linearized equation, so long as
ε
>
0
is sufficiently small. Additionally, the marginals
∫
f dv and
∫
f dx converge exponentially fast to their equilibrium value, in all C
^r
spaces.

All the estimates appearing in the nonlinear statement are constructive.

Clément Mouhot