Ducomet, Bernard; Nečasová, Šárka; Vasseur, Alexis. On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities. Z. Angew. Math. Phys. 61 (2010), no. 3, 479--491.
By Eq. (12), we see readily that the authors concerns about the finite mass case, and thus $$\bex \int \rho\rd x\leq C. \eex$$ Moreover, $$\bex \int \rho \ln \rho =\int \rho \frac{1}{\ve}\ln \rho^\ve \leq \int \frac{1}{\ve}\rho (\rho^\ve-1) \leq \frac{1}{\ve} \rho^{1+\ve},\quad\forall\ \ve>0, \eex$$ 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 $$\beex \bea \int \rho \ln \rho &\leq \frac{2}{\gm-1}\int \rho^\frac{1+\gm}{2}\\ &=\frac{2}{\gm-1}\int \frac{1}{\delta} \rho^\frac{1}{2}\cdot \delta \rho^\frac{\gm}{2}\\ &\leq \frac{1}{\gm-1}\int \sex{\frac{1}{\delta^2} \rho+\delta^2\rho^\gm}\\ &\leq C+\frac{\delta^2}{\gm-1}\int \rho^\gm,\quad \forall\delta>0. \eea \eeex$$ Choosing $\delta$ sufficiently small, we can then absorb the term $\int \rho \ln \rho$.
Remark. In the above calculations, only the restriction $\gm>1$ was used!
