A remark on the velocity averaging lemma of the transport equation with general case

Abstract

<abstract><p>In this paper, we proved a new result for the celebrated velocity averaging lemma of the free transport equation with general case</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \partial_{t}f+ a(v) \cdot \nabla_{x} f = 0\,. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>After averaging with some weight functions $ \varphi(v) $, we proved that the average quantity $ \rho_{\varphi}(t, x) = \int_{\mathbb{R}_{v}^{3}}f(t, x, v)\, \varphi(v)\, {\rm d} v $ is in $ W_{x}^{1, p} $, $ p\in[1, +\infty] $. This result revealed the regularizing effect for the mean value with respect to the velocity of the solution. Our strategy was taking advantage of a modified vector field method to build up a bridge between the $ x $-derivative and $ v $-derivative. One significant point was that we first observed that the operator $ t\, \nabla_{x}+\left(\left[ \nabla _{v} a(v) \right] ^{T}\right) ^{-1}\nabla_{v} $ commuted with $ \partial_{t}+ a(v) \cdot \nabla_{x} $.</p></abstract>