On the stability of rarefaction wave solutions for viscous \(p\)-system with boundary effect
This paper concerns the large time behavior of solutions of the isentropic compressible Navier-Stokes equations with a \(\gamma\)-law (\(\gamma\geq 1\)) in one space dimension, when the spatial domain is a half-line \(\{\,x>0\,\}\), and with boundary data corresponding to supersonic inflow (that is, the velocity of the fluid in the direction \(x\) is positive, and greater than the sound speed). From the hyperbolic point of view, supersonic inflow means that the two characteristics enter the domain: therefore, the hyperbolic Initial Boundary Value Problem belongs to a rather simple class in which both the density and the velocity should be prescribed at \(x=0\), as is done in the viscous case. This suggests that no boundary layer should be present in the viscous solutions. And this is indeed what is found in the paper under review: viscous solutions with fixed endstates (at \(x=0\) and \(x=+\infty\)) are shown to be asymptotic (when time goes to infinity) to inviscid smooth solutions (rarefaction waves). The proof relies on a reformulation of equations in Lagrangian coordinates and on elementary \(L^2\) \textit{a priori} estimates. The subsonic inflow problem (in which we do expect a boundary layer, because of an outgoing characteristic in the inviscid problem) was studied in an earlier work by \textit{A. Matsumura} and \textit{K. Nishihara} [Commun. Math. Phys. 222, No. 3, 449--474 (2001; Zbl 1018.76038)].
scientific article; zbMATH DE number 2101821
scientific article