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Global solutions to a one-dimensional non-conservative two-phase model

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  • In this paper we investigate a basic one-dimensional viscous gas-liquid model based on the two-fluid model formulation. The gas is modeled as a polytropic gas whereas liquid is assumed to be incompressible. A main challenge with this model is the appearance of a non-conservative pressure term which possibly also blows up at transition to single-phase liquid flow (due to incompressible liquid). We investigate the model both in a finite domain (initial-boundary value problem) and in the whole space (Cauchy problem). We demonstrate that under appropriate smallness conditions on initial data we can obtain time-independent estimates which allow us to show existence and uniqueness of regular solutions as well as to gain insight into the long-time behavior of the model. These results rely strongly on the fact that we can derive appropriate upper and lower uniform bounds on the gas and liquid mass. In particular, the estimates guarantee that gas does not vanish at any point for any time when initial gas phase has a positive lower limit. The discussion of the Cauchy problem is general enough to take into account the possibility that the liquid phase may vanish at some points at initial time.
    Mathematics Subject Classification: Primary: 76T10, 76N10; Secondary: 65M12, 35L60.

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