Velocity averaging -- a general framework

Martin Lazar; Darko Mitrovic

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arXiv · 1107.2616

Velocity averaging -- a general framework

Martin Lazar·Darko Mitrovic

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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.

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Martin Lazar, Darko Mitrovic. 2011-07-13. Velocity averaging -- a general framework. https://arxiv.org/abs/1107.2616

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