Velocity averaging, kinetic formulations, and regularizing effects in quasi‐linear PDEs

We prove in this paper new velocity‐averaging results for second‐order multidimensional equations of the general form ℒ︁(∇x,v)f(x, v) =g(x, v) where ℒ︁(∇x,v) :=a(v) · ∇x− ∇·b(v)∇x. These results quantify the Sobolev regularity of the averages, ∫vf(x, v)ϕ(v)dv, in terms of the nondegeneracy of the set {v: |ℒ︁(iξ,v)| ≤ δ} and the mere integrability of the data, (f, g) ∈ (L ,L ). Velocity averaging is then used to study theregularizing effectin quasi‐linear second‐order equations, ℒ︁(∇x, ρ)ρ =S(ρ), which use their underlying kinetic formulations, ℒ︁(∇x, v)χρ=gS. 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, nonisotropic diffusion.