Microlocal partition of energy for fractional-type dispersive equations

Yang, Haocheng

doi:10.2140/apde.2025.18.2081

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Microlocal partition of energy for fractional-type dispersive equations

Haocheng Yang

Vol. 18 (2025), No. 9, 2081–2143

DOI: 10.2140/apde.2025.18.2081

Abstract

This paper is devoted to the proof of the microlocal partition of energy for fractional-type dispersive equations including the Schrödinger equation, the linearized gravity or capillary water-wave equation and the half-Klein–Gordon equation. Roughly speaking, a quarter of the $L^{2}$ energy lies inside or outside the “light cone” $| x | = | t P^{'} (ξ) |$ for large time. In addition, based on the study of the half-Klein–Gordon equation, the microlocal partition of energy will also be proved for the Klein–Gordon equation.

Keywords

microlocal energy estimates, asymptotic behavior, boundedness of pseudodifferential operators, dispersive equations

Mathematical Subject Classification

Primary: 35B40

Secondary: 47G30, 76B15

Milestones

Received: 22 October 2023

Accepted: 29 October 2024

Published: 5 September 2025

Authors

Haocheng Yang
École Normale Supŕieure Paris-Saclay CNRS Centre Borelli UMR9010 Gif-sur-Yvette France	Université Paris XIII (Sorbonne Paris-Nord) LAGA, CNRS (UMR 7539) Villetaneuse France

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Vol. 18, No. 9, 2025