Global Strong Solution to a Thermodynamic Compressible Diffuse Interface
Model with Temperature Dependent Heat-conductivity in 1-D
Abstract
In this paper, we investigate the wellposedness of the non-isentropic
compressible Navier-Stokes/Allen-Cahn system with the heat-conductivity
proportional to a positive power of the temperature. This system
describes the flow of a two-phase immiscible heat-conducting viscous
compressible mixture. The phases are allowed to shrink or grow due to
changes of density in the fluid and incorporates their transport with
the current. We established the global existence and uniqueness of
strong solutions for this system in 1-D, which means no phase
separation, vacuum, shock wave, mass or heat or phase concentration will
be developed in finite time, although the motion of the two-phase
immiscible flow has large oscillations and the interaction between the
hydrodynamic and phase-field effects is complex. Our result can be
regarded as a natural generalization of the Kazhikhov-Shelukhin’s result
([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]) for the
compressible single-phase flow with constant heat conductivity to the
non-isentropic compressible immiscible two-phase flow with degenerate
and nonlinear heat conductivity.