In the article "Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity" whose doi is 10.1080/03605302.2010.516785
$(\rho_\varepsilon,u_\varepsilon)$ satisfy the approximation equations. (3.1)
$(\bar\rho,\bar u)$ is the rarefaction wave which satisfy the following equations.(2.3)(2.4)
we have the etismate in lemma 3.1
In the article it says in order to pass the limit $\varepsilon\to0$ we need the following higher estimates on the momentum.
By passing the limit we get the global weak solution $(\rho,u)$ in theorem 2.1,and the definition of weak solution is in definition 2.1
My question is how it pass the limit?
Is it true that $\rho_\varepsilon$ weak star converge to $\rho$ under subsequence meanning because that in lemma 3.2 $\rho_\varepsilon<C$ where $C$ is independent of $\varepsilon$?
But what is $u$, how to get the limit $u$?