Velocity averaging -- a general framework

TL;DR

This paper establishes a general framework for velocity averaging, proving strong precompactness of averaged solutions to variable coefficient differential equations, including hyperbolic, parabolic, and fractional types, with results depending on coefficient regularity.

Contribution

It introduces a unified approach to velocity averaging for various differential operators with variable coefficients, extending previous results to fractional and discontinuous coefficient cases.

Findings

Strong precompactness of averaged quantities in L^2 spaces.

Applicable to hyperbolic, parabolic, ultraparabolic, and fractional differential operators.

Handles discontinuous coefficients when s > 2.

Abstract

We prove that the sequence of averaged quantities $\int_{\R^m}u_n(\mx,\msnop)$ $\rho(\msnop)d\msnop$, is strongly precompact in $\Ldl\Rd$, where $\rho\in \Ldc{\R^m}$, and $u_n\in \Ld{\R^m; \pL s\Rd}$, $s\geq 2$, are weak solutions to differential operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators. If $s>2$ then the coefficients can be discontinuous with respect to the space variable $\mx\in \R^d$, otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the $H$-measures.