On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

Abstract

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

The maximal regularity and its application to a multi-dimensional non-conservative viscous compressible two-fluid model with capillarity effects in $L^{ p}$-type framework

“…for an integer l ≥ 3 are small enough. In L 2 -type critical Besov spaces, Xu et al [31] constructed the well-posedness and decay rates of strong solutions to a multi-dimensional system (1.1) without capillarity effects.…”