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

Ming-Jiea Lyu; Baoyan Sun; Ming-Jiea Lyu; Baoyan Sun

doi:10.3934/nhm.2024007

Networks and Heterogeneous Media

2024, Volume 19, Issue 1: 157-168. doi: 10.3934/nhm.2024007

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Research article

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

Ming-Jiea Lyu ¹,
Baoyan Sun ^{2
,
,}

1.
Department of Applied Mathematics, Chung Yuan Christian University, Taoyuan, Taiwan
2.
School of Mathematics and Information Sciences, Yantai University, Yantai, China

Received: 18 October 2023 Revised: 08 January 2024 Accepted: 25 January 2024 Published: 01 February 2024

In this paper, we proved a new result for the celebrated velocity averaging lemma of the free transport equation with general case

$ \begin{equation*} \partial_{t}f+ a(v) \cdot \nabla_{x} f = 0\,. \end{equation*} $

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} $.
- velocity averaging lemma,
- regularity,
- vector field method,
- nonrelativistic transport equation,
- relativistic transport equation
Citation: Ming-Jiea Lyu, Baoyan Sun. A remark on the velocity averaging lemma of the transport equation with general case[J]. Networks and Heterogeneous Media, 2024, 19(1): 157-168. doi: 10.3934/nhm.2024007

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Abstract

In this paper, we proved a new result for the celebrated velocity averaging lemma of the free transport equation with general case

$ \begin{equation*} \partial_{t}f+ a(v) \cdot \nabla_{x} f = 0\,. \end{equation*} $

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} $.

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