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In this paper, we study a system of PDE's describing the motion of twocompressible viscous fluids occupying the whole space $\mathbb {R}^d\;(d\in\{2,3\}$). We prove a local well-posedness result with discontinuous initialdata: we require the initial density to be H\"older continuous on both sides ofa hyper-surface across which it can experience large jumps. The initialvelocity belongs to the Sobolev space $H^1(\mathbb{R}^d)$ and the divergence ofthe initial stress tensor to belong to $L^2(\mathbb{R}^d)$. The laterassumption expresses somehow the continuity of the stress tensor. This resultis more general than the one by Tani [26] by allowing low regular initial data,and it can be used to serve as building blocks for global solutions.

A well-posedness result for the compressible two-fluid model with density-dependent viscosity