Loeper, Grégoire Uniqueness of the solution to the Vlasov–Poisson system with bounded density. (English) Zbl 1111.35045 J. Math. Pures Appl. (9) 86, No. 1, 68-79 (2006). The Vlasov-Poisson system describes the evolution of a cloud of electrons or gravitational matter by the equations \[ \partial_t f+\xi\cdot\nabla_xf-\nabla\Psi\cdot\nabla_\xi f=0,\quad -\Delta\psi=\varepsilon\rho \]where \(\rho(t,x)=\int f(t,x,\xi)\,d\xi\) and \(\varepsilon>0\) in the electrostatic (repulsive) case, \(\varepsilon<0\) in the gravitational (attractive) case. Here \[ f(t,x,\xi)\geq 0 \]denotes the density of electrons (or matter) at time \(t\in \mathbb{R}^+\), position \(x\in\mathbb{R}^3\), velocity \(\xi\in\mathbb{R}^3\). The second equation is to be understood in the following sense\[ \Psi(t,x)=\varepsilon\int_{\mathbb{R}^3}\rho(t,y)\frac{1}{4\pi| x-y| }dy. \]There is shown the uniqueness of weak solutions of this system under the only condition that the macroscopic density \(\rho(t,x)\) is bounded in \(L^{\infty}\). Reviewer: Boris V. Loginov (Ul’yanovsk) Cited in 5 ReviewsCited in 114 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 82D10 Statistical mechanics of plasmas 82C70 Transport processes in time-dependent statistical mechanics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow Keywords:Vlasov-Poisson system; optimal transportation; transport equations; uniqueness of weak solutions PDFBibTeX XMLCite \textit{G. Loeper}, J. Math. Pures Appl. (9) 86, No. 1, 68--79 (2006; Zbl 1111.35045) Full Text: DOI arXiv References: [1] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric space and in the Wasserstein space of probability measures, Birkhäuser, in preparation; L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric space and in the Wasserstein space of probability measures, Birkhäuser, in preparation [2] Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44, 4, 375-417 (1991) · Zbl 0738.46011 [3] DiPerna, R. J.; Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 3, 511-547 (1989) · Zbl 0696.34049 [4] Gangbo, W.; McCann, R. J., The geometry of optimal transportation, Acta Math., 177, 2, 113-161 (1996) · Zbl 0887.49017 [5] Lions, P.-L.; Perthame, B., Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105, 2, 415-430 (1991) · Zbl 0741.35061 [6] Loeper, G., A fully non-linear version of the Euler incompressible equations: the semi-geostrophic equations, Submitted preprint, available on arXiv and on · Zbl 1134.35046 [7] Majda, A. J.; Bertozzi, A. L., Vorticity and Incompressible Flow, Cambridge Texts Appl. Math., vol. 27 (2002), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0983.76001 [8] McCann, R. J., A convexity principle for interacting gases, Adv. Math., 128, 1, 153-179 (1997) · Zbl 0901.49012 [9] Robert, R., Unicité de la solution faible à support compact de l’équation de Vlasov-Poisson, C. R. Acad. Sci. Paris Sér. I Math., 324, 8, 873-877 (1997) · Zbl 0886.35118 [10] Villani, C., Topics in Optimal Transportation, Grad. Stud. Math., vol. 58 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1106.90001 [11] Youdovitch, V., Non-stationary flows of an ideal incompressible fluid, Zh. Vych. Mat., 3, 1032-1066 (1963) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.