Existence and stability of weak solutions of the Vlasov-Poisson system in localised Yudovich spaces

We consider the Vlasov-Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. Our first main theorem yields the analog for the Vlasov-Poisson system of Yudovich’s celebrated well-posedness theorem for the Euler equations: we prove the uniqueness and the quantitative stability of Lagrangian solutions f = f ( t , x , v ) whose associated spatial density ρ f = ρ f ( t , x ) is potentially unbounded but belongs to suitable uniformly-localised Yudovich spaces. This requirement imposes a condition of slow growth on the function p ↦ ‖ ρ f ( t , ⋅ ) ‖ L p uniformly in time. Previous works by Loeper, Miot and Holding-Miot have addressed the cases of bounded spatial density, i.e. ‖ ρ f ( t , ⋅ ) ‖ L p ≲ 1 , and spatial density such that ‖ ρ f ( t , ⋅ ) ‖ L p ∼ p 1 / α for α ∈ [ 1 , + ∞ ) . 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.