Stability of a compressible two-fluid hyperbolic-elliptic system arising in fluid mechanics

This paper deals with an initial-boundary value problem for the following one-dimensional two-fluid system {nt+(nug)x=0,x∈I=(0,1),t>0,mt+(mul)x=0,αg(Pg)x=μg(ug)xx,αl(Pl)x=μl(ul)xx,αl+αg=1, where n and m represent, respectively, gas mass and liquid mass; ug and ul are corresponding fluid velocities whereas αg and αl are volume fractions occupied by the gas and liquid phase, and Pg and Pl are pressures associated with them. The model represents a submodel of the full two-fluid model studied in Bresch et al. (2012). An important difference between the model studied in the present work and that studied in Bresch et al. (2012) is that viscosity coefficients μl,μg are assumed to be constant. Bresch et al. assumed mass-dependent coefficients that allowed them to derive a so-called BD inequality which implies that masses are in H1. Since we are excluded from following that route, we instead explore how the use of two non-equal pressure functions Pg and Pl (i.e., Pl−Pg=f(m)≠0) allows us to obtain global estimates that guarantee a stability result to hold. I.e., we prove that m(⋅,t)→m˜,n(⋅,t)→n˜,ul(⋅,t),ug(⋅,t)→0,ast→∞, with respect to the norm in L∞(I) for constant states m˜ and n˜. Estimates of the time asymptotic behavior are also provided.