[Mélange de phase pour l’équation de Hartree et amortissement Landau dans la limite semi-classique]
The asymptotic behaviour of the Hartree equation is studied near translation-invariant steady states. For short-range interaction kernels satisfying a uniform Penrose stability condition, including the screened Coulomb interaction, phase-mixing estimates in finite regularity are established. These demonstrate density decay and scattering of solutions in weighted quantum Sobolev spaces, providing a quantum analogue of Landau damping in classical plasma physics. The results hold uniformly in the semiclassical limit, thereby bridging the quantum and classical regimes.
Nous étudions le comportement asymptotique de l’équation de Hartree au voisinage d’états stationnaires invariants par translation. Pour des noyaux d’interaction à courte portée satisfaisant une condition de stabilité de Penrose uniforme, parmi lesquels l’interaction de Coulomb écrantée, nous établissons des estimations de mélange de phases en régularité finie. Ces estimations nous permettent de démontrer la décroissance de la perturbation de densité et le scattering des solutions dans des espaces de Sobolev quantiques à poids, fournissant ainsi un analogue quantique de l’amortissement Landau en physique classique des plasmas. Les résultats sont uniformes dans la limite semi-classique et établissent ainsi un lien entre les régimes quantique et classique.
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Keywords: Hartree equation, Landau damping, phase mixing, semiclassical limit, Penrose condition, Wigner transform, scattering
Mots-clés : Équation de Hartree, amortissement Landau, mélange de phase, limite semi-classique, condition de Penrose, transformée de Wigner, scattering.
Marnie Smith 1
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Marnie Smith. Phase mixing for the Hartree equation and Landau damping in the semiclassical limit. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1483-1546. doi: 10.5802/jep.353
@article{JEP_2026__13__1483_0,
author = {Marnie Smith},
title = {Phase mixing for the {Hartree} equation and {Landau} damping in the semiclassical limit},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1483--1546},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.353},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.353/}
}
TY - JOUR AU - Marnie Smith TI - Phase mixing for the Hartree equation and Landau damping in the semiclassical limit JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 1483 EP - 1546 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.353/ DO - 10.5802/jep.353 LA - en ID - JEP_2026__13__1483_0 ER -
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