Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier–Stokes equation

The paper proves time-asymptotic stability of a 1-rarefaction plus 2-viscous shock for 1D barotropic Navier–Stokes by a weighted relative-entropy method in BD effective variables U=(v,h) with a specially engineered weight a(ξ) and a shift ODE tailored to the shock layer; this yields a coercive “good” term after a change of variables and a sharp Poincaré inequality, allowing closure of estimates and yielding sup-norm convergence and Ẋ(t)→0. By contrast, the model’s outline works in the classical variables (v,u), relies on a generic a-weighted Euler relative entropy, and chooses the shift by an orthogonality to the shock translation mode. It omits the crucial use of effective velocity h and the specific shift ODE that activate the entropy flux’s good term; with (v,u) alone, the key contraction/compensation mechanism does not follow. It also mixes exact rarefaction with mentions of a smoothed profile but does not construct it or quantify the error terms precisely. Hence the paper’s argument is correct and complete, while the model’s solution misses essential structural ingredients and is not justified.