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

Abstract 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 pro...

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Bibliographic Details
Published in:Nonlinearity 2024-09, Vol.37 (9), p.95015
Main Authors: Crippa, Gianluca, Inversi, Marco, Saffirio, Chiara, Stefani, Giorgio
Format: Article
Language:English
Subjects:
34A12
Cauchy problem
Lagrangian stability
Osgood condition
Primary 35Q83
Secondary 82D10
Vlasov–Poisson equations
Yudovich spaces
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Summary:Abstract 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.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/ad5bb3