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

Steinar Evje; Huanyao  Wen; Lei Yao

doi:10.3934/dcds.2016.36.1927

Article Contents

2016, Volume 36, Issue 4: 1927-1955. Doi: 10.3934/dcds.2016.36.1927

This issue Previous Article On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications Next Article Real bounds and Lyapunov exponents

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

1.
University of Stavanger (UiS), 4036 Stavanger
2.
University of Stavanger, NO-4036 Stavanger
3.
School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127

Received: March 2014

Revised: November 2014

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 76T10, 76N10; Secondary: 65M12, 35L60.

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