engleski

Velocity averaging – a general framework

We prove that the sequence of averaged quantities $\int_{; ; ; ; ; R^m}; ; ; ; ; u_n(x, p)$ $\rho(p) dp$, is strongly precompact in $L^2_{; ; ; ; ; loc}; ; ; ; ; (R^d)$, where $\rho\in L^2_c(R^m)$, and $u_n\in L^2(R^m ; L^s(R^d))$, $s\geq 2$, 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 $x\in R^d$, otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the H-measures.

velocity averaging; generalised H-measures; ultraparabolic equations; discontinuous coefficients; entropy solutions.

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