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arXiv · 2609.08586

On the Invertibility of the Potential-Density Mapping for the Vlasov-Poisson System in Analytic Spaces

Simon Le Bouëdec
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Abstract

We develop the mathematical analysis of the Time-Dependent Density Functional Theory introduced by Manfredi [J. Plasma Phys., 86.2 (2020), 825860201] in the setting of Vlasov equations with a time-dependent exterior potential. In this context, we establish the invertibility of the potential-density mapping for the Vlasov-Poisson system on $\mathbb{T}^d \times \mathbb{R}^d$ when the exterior potential is (real-)analytic in the space variable. We also recover a Runge-Gross type Theorem for the Vlasov-Poisson equation for smooth (real-)analytic in time exterior potential. The surjectivity of the potential-density mapping is an inverse problem for the Vlasov-Poisson equation: we find the electric field that generates a given density profile on a given time interval.

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Simon Le Bouëdec. 2026-09-08. On the Invertibility of the Potential-Density Mapping for the Vlasov-Poisson System in Analytic Spaces. https://arxiv.org/abs/2609.08586

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