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On PDE analysis of flows of quasi‐incompressible fluids

We study mathematical properties of quasi‐incompressible fluids. These are mixtures in which the density depends on the concentration of one of their components. Assuming that the mixture meets mass and volume additivity constraints, this density‐concentration relationship is given explicitly. We show that such a constrained mixture can be written in the form similar to compressible Navier‐Stokes equations with a singular relation between the pressure and the density. This feature automatically leads to the density bounded from below and above. After addressing the choice of thermodynamically compatible boundary conditions, we establish the large data existence of weak solution to the relevant initial and boundary value problem. We then investigate one possible limit from the quasi‐compressible regime to the incompressible regime.