High-capillarity limit and smoothing effect of large solutions for a multi-dimensional generic non-conservative compressible two-fluid model

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(\alpha^{-}\rho^{-})=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)}{\alpha^{-}(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.