### Velocity Averaging, Kinetic Formulations and Regularizing Effects in Quasilinear PDEs

Abstract: We prove new velocity averaging results for second-order multidimensional equations of the general form, ${\mathcal L}(\nabla_x,v)f(x,v)=g(x,v)$ where ${\mathcal L}(\nabla_x,v):={\mathbf a}(v)\cdot\nabla_x-\nabla_x^\top\cdot{\mathbf b}(v)\nabla_x$. These results quantify the Sobolev regularity of the averages, $\int_vf(x,v)\phi(v)dv$, in terms of the non-degeneracy of the set $\{v\!: |{\mathcal L}(i\xi,v)|\leq \delta\}$ and the mere integrability of the data, $(f,g)\in (L^p_{x,v},L^q_{x,v})$. Velocity averaging is then used to study the \emph{regularizing effect} in quasilinear second-order equations, ${\mathcal L}(\nabla_x,\rho)\rho=S(\rho)$ using their underlying kinetic formulations, ${\mathcal L}(\nabla_x,v)\chi_\rho=g_{{}_S}$. 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.

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