On a Non-conservative Compressible Two-Fluid Model in a Bounded Domain: Global Existence and Uniqueness
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Journal of Mathematical Fluid Mechanics
On a Non-conservative Compressible Two-Fluid Model in a Bounded Domain: Global Existence and Uniqueness Yinghui Wang, Huanyao Wen and Lei Yao Communicated by G. P. Galdi
Abstract. In this work, we consider the Dirichlet problem for a non-conservative compressible two-fluid model in three dimensions. In particular, capillary pressure is taken into account in the sense that P + − P − = f = 0 where f is assumed to be a strictly decreasing function near the equilibrium. This work aims to prove that this assumption has an essential stabilization effect on the model in bounded domains. Mathematics Subject Classification. 76T10, 76N10, 35B40. Keywords. Compressible two-fluid model, Compressible Navier–Stokes equations, Global existence and uniqueness, Asymptotic behavior, Dirichlet problem.
1. Introduction 1.1. Background In this paper, we consider the following generic compressible two-fluid model in a bounded domain Ω ⊂ R3 : ⎧ + α + α− = 1, ⎪ ⎪ ⎪ ⎨ ∂t (α± ρ± ) + div(α± ρ± u± ) = 0, (1.1) ⎪ ∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(α± τ ± ), ⎪ ⎪ ⎩ + + P (ρ ) − P − (ρ− ) = f (α− ρ− ), where x ∈ Ω is the spatial variable, t is the time variable. 0 ≤ α+ (x, t) ≤ 1 is the volume fraction of fluid ± +, and 0 ≤ α− (x, t) ≤ 1 is that of fluid −. Moreover ρ± (x, t) ≥ 0, u± (x, t) and P ± (ρ± ) = A± (ρ± )γ¯ are the densities, velocities and pressures of each phase. It is assumed that γ¯ ± ≥ 1, and A± > 0 are constants. Without loss of generality, we set A+ = A− = 1 in the sequel. In addition, the function f (·) ∈ C 3 ([0, +∞)) is strictly decreasing near the equilibrium. Also, τ ± are the viscous stress tensors: τ ± := μ± (∇u± + ∇t u± ) + λ± divu± Id, where the constants μ± and λ± are shear and bulk viscosity coefficients satisfying μ± > 0, and 2μ± + 3λ± ≥ 0, which deduces μ± + λ± > 0. In (1.1), we describe the capillary pressure effects in terms of a pressure difference function f in (1.1)4 , which is commonly included in modeling of two-phase flow in porous media, see [12,13] and references therein. For more physical background about the model, we refer to [2,3,12,16,24] and references therein. 0123456789().: V,-vol
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Equal Velocities and Equal Pressure Functions When velocities, pressure functions are equal, i.e., u+ = u− and P + = P − , the system will become conservative form. In fact, this can be achieved by summing the two momentum equations together and using the fact that α+ + α− = 1. In this case, some good properties of conservation laws are helpful, for instance, the derivative of the pressure can be shifted to a test function such that it is not necessary for the solution space to include derivative of density. This yields more possibilities to obtain global solutions with arbitrarily large data. In this context, some global existence results allowing large initial data have been obtained. For the one-dimensional case, Evje and Karlsen [9] obtained the global existence of weak solution when two densities are bounded and away from
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