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

DOI10.1002/CPA.20180zbMATH Open1131.35004arXivmath/0511054OpenAlexW2140946313MaRDI QIDQ5310278

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Publication date: 21 September 2007

Published in: (Search for Journal in Brave)

Abstract: We prove new velocity averaging results for second-order multidimensional equations of the general form,

o p (a b l a_{x}, v) f (x, v) = g (x, v)

where

. These results quantify the Sobolev regularity of the averages,

i n t_{v} f (x, v) p h i (v) d v

, in terms of the non-degeneracy of the set

v : | o p (i x i, v) | l e q d e l t a

and the mere integrability of the data,

(f, g) i n (L_{x, v}^{p}, L_{x, v}^{q})

. Velocity averaging is then used to study the emph{regularizing effect} in quasilinear second-order equations,

o p (a b l a_{x}, h o) h o = S (h o)

using their underlying kinetic formulations,

o p (a b l a_{x}, v) c h i_{h} o = 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.

Full work available at URL: https://arxiv.org/abs/math/0511054

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