Preprint 2005-044

Velocity Averaging, Kinetic Formulations and Regularizing Effects in Quasilinear PDEs

Eitan Tadmor and Terence Tao

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.

Available as PDF (400 Kbytes).
Eitan Tadmor, email: tadmor at cscamm dot umd dot edu
Terence Tao, email: tao at math dot ucla dot edu
Publishing information:
Submitted by:
tadmor at cscamm dot umd dot edu November 3 2005.

[ 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | All Preprints | Preprint Server Homepage ]
© The copyright for the following documents lies with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use.

Conservation Laws Preprint Server <>
Last modified: Mon Nov 7 13:25:59 MET 2005