Singular and regular analysis for the free boundaries of two-phase inviscid fluids in gravity field
Lili Du, Feng Ji
TL;DR
The paper advances the theory of two-phase free boundary problems under gravity by establishing a rigorous blow-up framework that couples monotonicity formulas and frequency analysis to classify stagnation-point behavior. For the partial-degenerate regime (λ>0), it yields a half-plane limit for the decaying negative phase and a Stokes-type 120° corner for the positive phase in detached configurations, with symmetry broken in overlapping configurations; for the complete-degenerate regime (λ=0), it rules out two-phase stagnation points, reducing the local picture to one-phase Stokes corners. The results extend the Stokes conjecture to two-phase gravity flows and use concentration-compactness to handle nontrivial cavity structures, weighted densities, and the interaction between the two interfaces. Overall, the work provides a comprehensive regularity and asymptotic profile theory for two-phase free boundaries in gravity fields, clarifying how gravity and interfacial coupling shape stagnation-point singularities. The combination of variational methods, Weiss-type energies, blow-up analysis, and a dedicated frequency framework yields a robust classification of singularities and confirms symmetry and regularity properties of the free boundaries under the gravity-affected two-phase dynamics.
Abstract
In this paper, we consider a free boundary problem of two-phase inviscid incompressible fluid in gravity field. The presence of the gravity field induces novel phenomena that there might be some stagnation points on free surface of the two-phase flow, where the velocity field of the fluid vanishes. From the mathematical point of view, the gradient of the stream function degenerates near the stagnation point, leading to singular behaviors on the free surface. The primary objective of this study is to investigate the singularity and regularity of the two-phase free surface, considering their mutual interaction between the two incompressible fluids in two dimensions. More precisely, if the two fluids meet locally at a single point, referred to as the possible two-phase stagnation point, we demonstrate that the singular side of the two-phase free surface exhibits a symmetric Stokes singular profile, while the regular side near this point maintains the $C^{1,α}$ regularity. On the other hand, if the free surfaces of the two fluids stick together and have non-trivial overlapping common boundary at the stagnation point, then the interaction between the two fluids will break the symmetry of the Stokes corner profile, which is attached to the $C^{1,α}$ regular free surface on the other side. As a byproduct of our analysis, it's shown that the velocity field for the two fluids cannot vanish simultaneously on the two-phase free boundary. Our results generalize the significant works on the Stokes conjecture in [V$\check{a}$rv$\check{a}$ruc$\check{a}$-Weiss, Acta Math., 206, (2011)] for one-phase gravity water wave, and on regular results on the free boundaries in [De Philippis-Spolaor-Velichkov, Invent. Math., 225, (2021)] for two-phase fluids without gravity.