The Averaging lemma and regularizing effect
Abstract
We prove new velocity averaging results for second-order multidimensional equations of the general form, (∇x,v)f(x,v)=g(x,v) where (∇x,v):=(v)·∇x-∇x·(v)∇x. These results quantify the Sobolev regularity of the averages, ∫vf(x,v)φ(v)dv, in terms of the non-degeneracy of the set \v: |(,v)|≤ δ\ and the mere integrability of the data, (f,g)∈ (Lpx,v,Lqx,v). Velocity averaging is then used to study the regularizing effect in quasilinear second-order equations, (∇x,)=S() using their underlying kinetic formulations, (∇x,v)=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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