Microlocal Partition of Energy for Fractional-Type Dispersive Equations
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 L2 energy lies inside or outside the "light cone" |x| = |tP'()| 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.
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