Local Well-Posedness for Vlasov--Poisson with L^d+ Initial Density and Fractional Velocity Regularity

2026/07/27 by Quoc-Hung Nguyen
#math.AP

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Abstract

We prove a local well-posedness criterion for the Vlasov--Poisson system on \R^d_x×\R^d_v, d≥2, under an anisotropic assumption on the initial distribution. The datum has finite mass, its weighted velocity supremum belongs to L^p_x for some p>d, and it has an arbitrarily small positive Hölder regularity in the velocity variable, uniformly with respect to velocity and with the same spatial L^p control. The main estimate is a nonlinear mixing bound [ρ(t)]_C^αx\lesssim t^-d/p-εC(f₀), ε>0~~small. Thus the density is integrable in time with values in a positive spatial Hölder class, and the corresponding electric field belongs to L¹_tC^1,α_x. We construct a solution by a Schauder fixed point on the density and prove uniqueness by a Loeper-type stability estimate.

Citations

• Low regularity Sobolev well-posedness for Vlasov--Poisson
• Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity
• Existence and stability of weak solutions of the Vlasov--Poisson system in localized Yudovich spaces
• Stability estimates for the Vlasov-Poisson system in p-kinetic Wasserstein distances
• Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space
• Uniqueness and stability for the Vlasov-Poisson system with spatial density in Orlicz spaces
• A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system
• Potential estimates and quasilinear parabolic equations with measure data
• Uniqueness of the solution to the Vlasov-Poisson system with bounded density

Related

• Low regularity Sobolev well-posedness for Vlasov--Poisson
• Existence and stability of weak solutions of the Vlasov--Poisson system in localized Yudovich spaces
• Global well-posedness of Vlasov-Poisson-Boltzmann equations with neutral initial data and small relative entropy
• Gevrey regularity for the Vlasov-Poisson system
• The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness