Ducomet, Bernard; Ne?asová,šárka; Vasseur, Alexis. On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities. Z. Angew. Math. Phys. (+), No. 3, 479--491.
by Eq. (+), we see readily that the authors concerns about the finite mass case, and thus $$\bex \int, \rho\rd x\leq C. \e ex$$ Moreover, $$\bex \int \rho \ln \rho =\int \rho \frac{1}{\ve}\ln \rho^\ve \leq \int \frac{1}{\ve}\rho (\rho^\ve-1) \le Q \frac{1}{\ve} \rho^{1+\ve},\quad\forall\ \ve>0, \eex$$ where we have used the following fundamental inequality $$\bex \ln x\leq x-1,\quad \forall\ x>0. \eex$$ taking $\ve=\frac{\gm-1}{2}$, we have $$\beex \bea \int \rho \ln \rho &\leq \frac{2}{\gm-1}\int \rho^\frac{1+\g m}{2}\\ &=\frac{2}{\gm-1}\int \frac{1}{\delta} \rho^\frac{1}{2}\cdot \delta \rho^\frac{\gm}{2}\\ &\leq \frac{ 1}{\gm-1}\int \sex{\frac{1}{\delta^2} \rho+\delta^2\rho^\gm}\\ &\leq c+\frac{\delta^2}{\gm-1}\int \rho^\gm,\ Quad \forall\delta>0. \eea \eeex$$ Choosing $\delta$ sufficiently small, we can then absorb the term $\int \rho \ln \rho$.
Remark. In the above calculations, only the restriction $\gm>1$ is used!
A note to "on global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities"