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 ${\mathrm{L}}^p({\mathbb{R}}^{d+m})$, with $p>1$. We prove that the sequence of averaged quantities $(\int ρ(\my) u_n(\mx,\my) d\my)$ is strongly precompact in $\Ljl\Rd$ for any $ρ\in \Cc{\R^m}$, 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 $\LL p$, $p\ge 2$, framework under the standard non-degeneracy conditions, as well as a connection between the H-measures and the H-distributions.

New results concerning $L^s$-velocity averagingm $s\ge 2$ for equations with optimal eough coefficients are added

Cited by in corpus (1)

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