Stability of a compressible two-fluid hyperbolic–elliptic system arising in fluid mechanics. (October 2016) - Evje, Steinar; Wen, Huanyao
Abstract: This paper deals with an initial–boundary value problem for the following one-dimensional two-fluid system n t + ( n u g ) x = 0, x ∈ I = ( 0, 1 ), t > 0, m t + ( m u l ) x = 0, α g ( P g ) x = μ g ( u g ) x x, α l ( P l ) x = μ l ( u l ) x x, α l + α g = 1, where n and m represent, respectively, gas mass and liquid mass; u g and u l are corresponding fluid velocities whereas α g and α l are volume fractions occupied by the gas and liquid phase, and P g and P l 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 μ l, μ g 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 H 1 . Since we are excluded from following that route, we instead explore how the use of two non-equal pressure functions P g and P l (i.e., P l − P g = 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 ˜, u l ( ⋅, t ), u g ( ⋅, t ) → 0, as t → ∞, with respect to the norm in L ∞ ( I ) for constant states m ˜ and n ˜ . Estimates of the time asymptotic behavior are also provided.,