On the velocity averaging for equations with optimal heterogeneous rough coefficients

Abstract

Assume that $(u_n)$ is a sequence of solutions to heterogeneous equations with rough coefficients and fractional derivatives, weakly converging to zero in ${\rm L}^p(\R^{d+m})$, with $p>1$. We prove that the sequence of averaged quantities $(\int \rho(\my) u_n(\mx,\my) d\my)$ is strongly precompact in $\Ljl\Rd$ for any $\rho\in \Cc{\R^m}$, provided that restrictive non-degeneracy conditions are satisfied. These are fulfilled for elliptic, parabolic, fractional convection-diffusion equations, as well as for parabolic equations with a fractional time derivative. The main tool that we are using is an adapted version of H-distributions. As a consequence of the introduced methods, we obtain an optimal velocity averaging result in the $\LL p$, $p\geq 2$, framework under the standard non-degeneracy conditions, as well as a connection between the H-measures and the H-distributions.