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The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension N ⩾ 2. We first study the unique global solvability of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. Due to the presence of the capillary terms, we exploit the parabolic properties of the linearized system for all frequencies which enables us to apply contraction mapping principle to show the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the optimal time decay rates for the constructed global solutions.

In this paper, we revisit the optimal time decay rates of classical solutions to the 3D compressible Navier‐Stokes equations. Based on a global priori estimate, the optimal time decay rates of the solution and its first order derivative in L2‐norm are obtained if Hl‐norm(l≥4) of the initial perturbation around a constant state is small enough and false∥false(ρ0−trueρ¯,m0false)false∥Ḃ1,∞−sℓ is bounded for s∈[0,12). Compared with the previous works by Hai‐Liang Li and Ting Zhang (Math Meth Appl Sci 34:670‐682, 2011), we remove the smallness of false∥false(ρ0−trueρ¯,m0false)false∥Ḃ1,∞−s and our condition involves only the low frequencies of the data in trueḂ1,∞−sfalse(R3false).

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The present paper deals with the Cauchy problem of a multidimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality. 1. Introduction. It is well known that models of two-phase or multiphase flows are widely applied to study the hydrodynamics in industry, for example, in manufacturing, engineering, and biomedicine, where the fluids under investigation contain more than one component. In fact, it has been estimated that over half of everything produced in a modern industrial society depends, to some degree, on a multiphase flow process for its optimum design and safe operation. In nature, there is a variety of different multiphase flow phenomena, such as sediment transport, geysers, volcanic eruptions, clouds, and rain [2, 4]. In addition, models of multiphase flows also naturally appear in many contexts within biology, ranging from tumor biology and anticancer therapies to developmental biology and plant physiology[17]. The principles of single-phase flow fluid dynamics and heat transfer are relatively well