engleski

Velocity averaging - a general framework

We establish the strong $\Ldl\Rd$ precompactness of the sequence of averaged quantities $\int_{; ; \R^m}; ; u_n(\mx, \msnop)$ $\rho(\msnop)d\msnop$, 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 $\mx\in \R^d$. The main tool in the work is an extension of the H-measures, for which a representation theorem is proved. An application is give n to ultraparabolic equations with discontinuous coefficients.

velocity averaging; generalised H-measures; ultraparabolic equations; discontinuous coefficients

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