A Regularized Framework and Admissible Solutions for Liquid-Vapor Phase Transitions in Steady Compressible Flows

Abstract

We investigate the well-posedness of the periodic boundary value problem for the steady compressible isentropic Navier-Stokes system under the van der Waals equation of state. The main difficulty arises from the non-monotonicity of the pressure, which induces liquid-vapor phase transitions and consequently leads to both physical instabilities and mathematical non-uniqueness of solutions. It is shown that the occurrence of a phase transition is determined by whether the integral average of the specific volume lies inside the gas-liquid coexistence region defined by the Maxwell construction. By introducing an artificial viscosity, we construct an approximate system. When the integral average of the specific volume falls within the Maxwell region, the approximate solution converges, as the artificial viscosity tends to zero, to the equilibrium states given by Maxwell's construction, with the diffuse interface sharpening into a discontinuity. Conversely, if the integral average of the specific volume lies outside this region, the limiting solution remains outside as well, meaning that no phase transition occurs. These results demonstrate that the non-monotonicity of the pressure, combined with the condition that the integral average of the specific volume belongs to the Maxwell region, can act as a nucleation mechanism for phase transitions in the isentropic gas-liquid problem. Furthermore, the proposed approximation not only offers a regularized framework for describing phase transitions but also provides, from a rigorous mathematical viewpoint, a definition of admissible solutions related to phase transitions. The detailed proof relies on the artificial viscosity method, the calculus of variations, the anti-derivative technique, phase-plane analysis, and the level-set method.