*Submitted Paper*

**Inserted:** 2 jun 2023

**Last Updated:** 9 aug 2023

**Year:** 2023

**Abstract:**

We consider the Vlasov-Poisson system both in the repulsive (electrostatic
potential) and in the attractive (gravitational potential) cases. In our first
main theorem, we prove the uniqueness and the quantitative stability of
Lagrangian solutions $f=f(t,x,v)$ whose associated spatial density
$\rho_f=\rho_f(t,x)$ is potentially unbounded but belongs to suitable
uniformly-localized Yudovich spaces. This requirement imposes a condition of
slow growth on the function $p \mapsto \

\rho_f(t,\cdot)\

_{L^p}$ uniformly in
time. Previous works by Loeper, Miot and Holding--Miot have addressed the cases
of bounded spatial density, i.e., $\

\rho_f(t,\cdot)\

_{L^p} \lesssim 1$, and
spatial density such that $\

\rho_f(t,\cdot)\

_{L^p} \sim p^{1/\alpha}$ for
$\alpha\in[1,+\infty)$. Our approach is Lagrangian and relies on an explicit
estimate of the modulus of continuity of the electric field and on a
second-order Osgood lemma. It also allows for iterated-logarithmic
perturbations of the linear growth condition. In our second main theorem, we
complement the aforementioned result by constructing solutions whose spatial
density sharply satisfies such iterated-logarithmic growth. Our approach relies
on real-variable techniques and extends the strategy developed for the Euler
equations by the first and fourth-named authors. It also allows for the
treatment of more general equations that share the same structure as the
Vlasov-Poisson system. Notably, the uniqueness result and the stability
estimates hold for both the classical and the relativistic Vlasov-Poisson
systems.

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