Moisture dynamics with phase changes coupled to heat-conducting, compressible fluids

TL;DR

This work rigorously analyzes a moisture-thermodynamics model coupled to non-isothermal, heat-conducting compressible Navier–Stokes equations, incorporating phase-change phenomena with latent heat. It develops a robust Lagrangian framework and an $L^p$–$L^q$ maximal-regularity theory for the linearized problem, then uses fixed-point arguments to establish local strong well-posedness for large data. For small data, near-equilibrium analysis combined with perturbation theory yields global strong well-posedness, leveraging a blend of maximal regularity and energy methods to handle non-differentiable phase-change terms. The results advance the mathematical foundation for fully coupled moist atmospheric flows with phase changes, providing rigorous guarantees essential for analysis and simulation of moist, non-isothermal atmospheres.

Abstract

It is shown that a model coupling the heat-conducting compressible Navier-Stokes equations to a micro-physics model of moisture in air is locally strongly well-posed for large data in suitable function spaces and strongly well-posed on $[0,τ]$ for every $τ> 0$ for small initial data. This seems to be the first result on $[0,τ]$ for arbitrary $τ> 0$ for a model coupling moisture dynamics to heat-conducting, compressible Navier-Stokes equations. A key feature of the micro-physics model is that it also includes phase changes of water in moist air. These phase changes are associated with large amounts of latent heat and thus result in a strong coupling to the thermodynamic equation. The well-posedness results are obtained by means of a Lagrangian approach, which allows to treat the hyperbolicity in the continuity equation. More precisely, optimal $\mathrm{L}^p$-$\mathrm{L}^q$ estimates are shown for the linearized system, leading to the local well-posedness result by a fixed point argument and suitable nonlinear estimates. For the well-posedness result on $[0,τ]$ for arbitrary $τ> 0$, a refined analysis of the linearized problem close to equilibria is carried out, and the roughness of the source term, induced by the phase changes, requires to establish delicate a priori bounds.

Theorems & Definitions (23)

• Theorem 3.1: Local strong well-posedness
• Theorem 3.2: Well-posedness on $[0,\tau]$ for large $\tau > 0$ for small data
• Remark 1
• Lemma 1
• Proposition 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6