Abstract
In this paper, we consider a compressible two-fluid system with a common velocity field and algebraic pressure closure in dimension one. Existence, uniqueness and stability of global weak solutions to this system are obtained with arbitrarily large initial data. Making use of the uniform-in-time bounds for the densities from above and below, exponential decay of weak solution to the unique steady state is obtained without any smallness restriction to the size of the initial data. In particular, our results show that degeneration to single-fluid motion will not occur as long as in the initial distribution both components are present at every point.
Highlights
In multi-component flows the presence of topologically complex interphase separating the components is a great difficulty from physical as well as mathematical point of view
In the present paper we immediately assume that the two components of the flow share a common velocity field and that their pressures are equal
It turns out that the idea can be adapted to our two-fluid model
Summary
In multi-component flows the presence of topologically complex interphase separating the components is a great difficulty from physical as well as mathematical point of view. The same model, but in semi-stationary Stokes regime, has been recently investigated by Bresch, Mucha and the third author in the three-dimensional setting They proved the global-in-time existence of weak solutions without any restriction on the initial data. Uniqueness and stability of weak solutions are obtained by making full use of the specific structure of the equations. Thanks to the structure of the pressure, we are able to adapt the argument from [26] so as to obtain the twosided bounds; see Lemma 4.1 Based on these bounds, we show the exponential decay of weak solution by choosing suitable test functions in the momentum equation and making another use of the structure of the pressure.
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