Un résultat de diffusion pour l’équation de Hartree autour de solutions non localisées

de Suzzoni, Anne-Sophie; Collot, Charles

Séminaire Laurent Schwartz — EDP et applications (2017-2018), Exposé no. 14, 12 p.

On s’intéresse à l’évolution d’un système de particules autour d’équilibres thermodynamiques présentant un nombre infini de particules. Il s’agit d’étudier la stabilité asymptotique de solutions à l’équilibre de l’équation de Hartree :

$$i{\partial }_{t}X=-△X+{w*𝔼\left(\right|X|}^2)X$$

où $X$ est un champ aléatoire, $w$ une fonction réelle qui caractérise les interactions entre particules, $*$ le produit de convolution et $𝔼$ est l’espérance. Cette équation admet des solutions dont les lois sont invariantes par translations temporelles et spatiales, elles sont donc non localisées. On exposera leur stabilité asymptotique à travers un résultat de diffusion.

We are interested in the evolution of a system of particles around thermodynamical equilibria presenting an infinite number of particles. That is, we study the asymptotic stability of solutions at equilibrium of the Hartree equation :

$$i{\partial }_{t}X=-△X+{w*𝔼\left(\right|X|}^2)X$$

where $X$ is a random field, $w$ is a real fonction that characterises the interactions between particles, $*$ is the convolution product, and $𝔼$ is the expectation. This equation admits solutions whose laws are invariant under temporal and spatial translations, they are thus nonlocalised. We will present their asymtotic stability through a scattering result.

[1] Claude Bardos, Laszlo Erdős, François Golse, Norbert Mauser, and Horng-Tzer Yau, Derivation of the Schrödinger-Poisson equation from the quantum $N$-body problem, C. R. Math. Acad. Sci. Paris 334 (2002), no. 6, 515–520. | DOI | Zbl

[2] Claude Bardos, François Golse, Alex D. Gottlieb, and Norbert J. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl. (9) 82 (2003), no. 6, 665–683. | DOI | MR | Zbl

[3] Niels Benedikter, Marcello Porta, and Benjamin Schlein, Mean–field evolution of fermionic systems, Communications in Mathematical Physics 331 (2014), no. 3, 1087–1131 (English). | DOI | MR | Zbl

[4] Thomas Chen, Younghun Hong, and Nataša Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Ration. Mech. Anal. 224 (2017), no. 1, 91–123. | DOI | MR | Zbl

[5] —, On the scattering problem for infinitely many fermions in dimensions $d\ge 3$ at positive temperature, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, 393–416. | DOI | MR | Zbl

[6] Anne-Sophie de Suzzoni, An equation on random variables and systems of fermions, , 2015. | arXiv

[7] Alexander Elgart, László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Nonlinear Hartree equation as the mean field limit of weakly coupled fermions, J. Math. Pures Appl. (9) 83 (2004), no. 10, 1241–1273. | DOI | MR | Zbl

[8] Jürg Fröhlich and Antti Knowles, A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction, J. Stat. Phys. 145 (2011), no. 1, 23–50. | DOI | MR | Zbl

[9] Zihua Guo, Zaher Hani, and Kenji Nakanishi, Scattering for the 3D Gross-Pitaevskii equation, Comm. Math. Phys. 359 (2018), no. 1, 265–295. | DOI | MR | Zbl

[10] Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Scattering for the Gross-Pitaevskii equation, Math. Res. Lett. 13 (2006), no. 2-3, 273–285. | DOI | MR | Zbl

[11] —, Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math. 11 (2009), no. 4, 657–707. | DOI | MR | Zbl

[12] Mathieu Lewin and Julien Sabin, The Hartree equation for infinitely many particles, II : Dispersion and scattering in 2D, Anal. PDE 7 (2014), no. 6, 1339–1363. | DOI | MR | Zbl

[13] —, The Hartree equation for infinitely many particles, I : Well-posedness theory, Comm. Math. Phys. 334 (2015), no. 1, 117–170. | DOI | MR | Zbl

[14] Jens Linhard, On the properties of a gas of charged particles, Dan. Vid. Selsk Mat.-Fys. Medd. 28 (1954), no. 1, 8.