High-capillarity limit and smoothing effect of large solutions for a multi-dimensional generic non-conservative compressible two-fluid model
Ling-Yun Shou, Jiayan Wu, Lei Yao, Yinghui Zhang
TL;DR
The paper analyzes a general multidimensional non-conservative compressible two-fluid model with capillarity in the high-capillarity regime. It develops a reformulation in terms of density fluctuations and employs a four-variable decoupling to reveal a two-phase Navier–Stokes–Korteweg structure, enabling a blend of dispersive and parabolic analysis. Under a sharp stability condition on the capillary pressure derivative and sufficiently large capillarity, the authors prove global existence of large solutions in critical Besov spaces, establish convergence to incompressible Navier–Stokes with explicit rates for ill-prepared data, and demonstrate optimal decay and smoothing for lower-regularity data. The results extend global well-posedness to large amplitudes without restrictive smallness assumptions and provide precise control of the incompressible limit, with a rigorous treatment of dispersion-diffusion interplay driven by capillarity. This work thus advances understanding of capillarity effects in multidimensional two-fluid flows and their long-time behavior.
Abstract
We investigate the global existence and long-time behavior of large solutions, in the high-capillarity regime, for a general multidimensional non-conservative compressible two-fluid model with the capillary pressure relation \(f(α^{-}ρ^{-})=P^{+}-P^{-}\). Our main contributions are threefold. First, for sufficiently large capillarity coefficients, we prove the existence and uniqueness of global solutions in critical Besov spaces for large initial perturbations, under the sharp stability condition \(-\frac{s_{-}^{2}(1,1)}{α^{-}(1,1)}<f^{\prime}(1)<0\), thereby removing the additional negativity restriction assumed by Evje--Wang--Wen [Arch. Ration. Mech. Anal. 221:1285--1316, 2016]. Second, we give a rigorous justification of the global-in-time convergence to the incompressible Navier-Stokes flows and obtain explicit convergence rates in critical spaces for ill-prepared data. Third, if in addition the initial perturbation lies in a lower-regularity Besov space, we derive optimal decay rates for the solution and for its derivatives of any order, revealing a long-term smoothing effect. To the best of our knowledge, this is the first result on global large-amplitude strong solutions for multidimensional compressible two-fluid flows. Our analysis exploits the interplay between dispersion (two-phase Gross--Pitaevskii structure) and parabolic dissipation, both induced by capillarity effects.