The Ricci-Bourguignon flow (R-B flow) is a general geometric evolving equation, which includes or relates to some famous geometric flows, for example the Ricci flow and the Yamabe flow, etc. In this paper we shall prove that for the R-B flow evolving on $[0,T)$, whose first eigenvalue ${\lambda }_0$ of the operator $-△+\frac{{(1-(n-1)\rho )}^2}{4(1-2(n-1)\rho )}R$ for the initial metric $\mathrm{g}(0)$ is positive, or $T>0$ is finite, an upper bound assumption of the scalar curvature implies a noncollapsing estimate of the volume, uniformly for all time. In order to derive this noncollapsing estimate, we firstly establish a logarithmic Sobolev inequality along the R-B flow, by using the monotone formula for the Perelman's functional, and then we can derive a Sobolev inequality along the R-B flow.

Ricci-Bourguignon流（RB流）是一个通用的几何演化方程，它包含或涉及一些著名的几何流，例如Ricci流和Yamabe流等。在本文中，我们将证明RB流的演化上 $[0，Ť）$，其第一个特征值 ${\lambda }_0$ 运营商 $-△+\frac{{（\mathrm{1个}-（ñ-\mathrm{1个}）\rho ）}^2}{4（\mathrm{1个}-2（ñ-\mathrm{1个}）\rho ）}\mathrm{[R}$ 初始指标 $\mathrm{G}（0）$ 是肯定的，或者 $Ť>0$如果是有限的，则标量曲率的上限假设意味着对体积的不塌陷估计，并且在所有时间内都是一致的。为了得出该非收敛估计，我们首先通过使用Perelman泛函的单调公式，沿RB流建立对数Sobolev不等式，然后可以沿RB流得出Sobolev不等式。