Uniqueness of the solution to the Vlasov-Poisson system with bounded density

By Grégoire Loeper. 2005

254 citations recorded by the source corpus.

Abstract In this paper, we show uniqueness of weak solutions to the Vlasov–Poisson system on the only condition that the macroscopic density ρ defined by ρ ( t , x ) = ∫ R d f ( t , x , ξ ) d ξ is bounded in L ∞ . Our proof is based on optimal transportation.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/182a06364b0585bee941d62e0f2f850658d80677

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In Applied Mathematics

Citations in the corpus

Moment Propagation for Weak Solutions to the Vlasov–Poisson System (2012, background): “• Some references for previously known uniqueness statements are [5, 6, 10, 11].”
On the Mean Field and Classical Limits of Quantum Mechanics (2015, background): “…of Dobrushin’s inequality (item (b) on the list above) sig-niﬁcantly simpliﬁes the original argument, and allows extending Dobrushin’s inequality to Monge-Kantorovich distances with arbitrary ﬁnite exponents (see [21] for the original argument for the Monge-Kantorovich distance with exponent 2).”
Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow (2011, methodology): “The technique used there is quite similar to the one of Loeper [18], who studied the Vlasov-Poisson equation and the Euler equation in vorticity formulation.”
On the Dynamics of Large Particle Systems in the Mean Field Limit (2013, background): “The interested reader is also referred to the very interesting paper [69] where MongeKantorovich distances with exponents different from 1 are used in the same context — see also [48].”
Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, methodology): “The Lagrangian method introduced by Loeper in [48] estimates the difference of weak solutions in the Wasserstein distance and is very useful for inviscid problems or problems with linear diffusion [47, 6, 24].”
Lp theory for the multidimensional aggregation equation (2011, background): “[34] G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math.”
Particles approximations of Vlasov equations with singular forces : Propagation of chaos (2011, methodology): “with an estimate on the growth of the support lets us use standard stability estimates to control the W1 distance of fN to another solution of the Vlasov equation, see Loeper’s result [43] recalled in Proposition 3.”
Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, unknown): “The most general uniqueness result for the Vlasov-Poisson system was obtained in [37].”
A review of the mean field limits for Vlasov equations (2014, background): “Strong solutions requires an initial data f 0 ∈ L1∩L∞(R2d) with compact support or enough moment in velocity (see [120] for the best uniqueness condition).”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “For instance the following lemma was both proven in [215] and later in [224] for different purposes.”