The Averaging lemma and regularizing effect

We prove new velocity averaging results for second-order multidimensional equations of the general form, $\text{op}({\mathrm{\nabla }}_x,v)f(x,v)=g(x,v)$ where $\text{op}({\mathrm{\nabla }}_x,v):=\text{bba}(v)\cdot {\mathrm{\nabla }}_x-{\mathrm{\nabla }}_x^{\mathrm{\top }}\cdot \text{bbb}(v){\mathrm{\nabla }}_x$. These results quantify the Sobolev regularity of the averages, ${\int }_{v}f(x,v)\varphi (v)dv$, in terms of the non-degeneracy of the set ${v: |\op(\ixi,v)|\leq \delta}$andthemereintegrabilityofthedata,$(f,g)\in (L^p_{x,v},L^q_{x,v})$. Velocity averaging is then used to study the \emph{regularizing effect} in quasilinear second-order equations, $\text{op}({\mathrm{\nabla }}_x,\rho )\rho =S(\rho )$ using their underlying kinetic formulations, $\text{op}({\mathrm{\nabla }}_x,v){\chi }_{\rho }=g_{{}_S}$. In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection-diffusion and elliptic equations driven by degenerate, non-isotropic diffusion.