Global weak solutions to a generic two-fluid model.

International audienceThe article discusses a two-fluid model. The fluids are described by their densities ρ±, velocities u± and the proportions α± in every point of space, α++α−=1. The underlying physical space is the 3-dimensional torus. The equations are ∂∂t(α±ρ±)+div(α±ρ±u±)=0, ∂∂t(α±ρ±u±)+div(α±ρ±u±⊗u±)+α±∇p=2μ±div(α±D(u±))+σ±α±ρ±∇Δ(α±ρ±), p=ργ. The constants μ±,σ± are the viscosity and surface tension coefficients. The article consists of two parts; in the first part the authors describe the theory of linearization of the equations. One of the conclusions of this part of their investigation is the following statement. Theorem. The region α−≥0 is invariant if and only if the fluid + is compressible. The second part concerns constructing global weak solutions for the nonlinear equations. The global weak solutions are constructed via a compactness argument. The compactness is established by proving a priori estimates using the conserved and monotonic quantities of the system. In addition to the standard mass, momentum and energy, the authors derive a new monotonic quantity with u±+ν∇R±R± where R±=α±ρ±. To construct approximate solutions, the authors add the following force to the momentum equations: εR±∇Δ2k+1R±−ε∇logR±−η(Δ2lu+u), where k,l are large integers and ε,η are parameters to be taken to zero to obtain the weak solution of the original system