Microlocal Partition of Energy for Fractional-Type Dispersive Equations

TL;DR

This paper establishes a microlocal partition of energy for fractional dispersive equations, showing that a quarter of the energy concentrates inside or outside the light cone, with extensions to Klein-Gordon equations.

Contribution

It introduces a microlocal partition of energy for fractional dispersive equations and extends the analysis to Klein-Gordon equations, providing new insights into energy distribution.

Findings

A quarter of the energy lies inside or outside the light cone for large times.

The microlocal partition of energy is proved for fractional dispersive equations.

Extension of the partition to Klein-Gordon equations based on half-Klein-Gordon analysis.

Abstract

This paper is devoted to the proof of microlocal partition of energy for fractional-type dispersive equations including Schr\"odinger equation, linearized gravity or capillary water-wave equation and half-Klein-Gordon equation. Roughly speaking, a quarter of the $L^2$ energy lies inside or outside the "light cone" $|x| = |tP'(\xi)|$ for large time. In addition, based on the study of half-Klein-Gordon equation, the microlocal partition of energy will also be proved for Klein-Gordon equation.