The Cauchy Problem for Non-Isentropic Compressible Navier-Stokes/Allen-Cahn system with Degenerate Heat-Conductivity

Abstract

The Cauchy problem for non-isentropic compressible Navier-Stokes/Allen-Cahn system with degenerate heat-conductivity \(\kappa (\theta) = \tilde \kappa {\theta ^\beta}\) in 1-D is discussed in this paper. This system is widely used to describe the motion of immiscible two-phase flow with diffused interface. The well-posedness for strong solution of this problem is established with the H¹ initial data for density, temperature, velocity, and the H² initial data for phase field. The result shows that no discontinuity of the phase field, vacuum, shock wave, mass or heat concentration will be developed at any finite time in the whole space. From the hydrodynamic point of view, this means that no matter how complex the interaction between the hydrodynamic and phase-field effects, phase separation will not occur, but the phase transition is possible.