Free Boundary Problem for inhomogeneous Navier-Stokes equations

TL;DR

This paper develops a new analytical framework for free boundary problems in inhomogeneous Navier-Stokes equations, addressing regularity and global solutions in critical spaces, with novel techniques for boundary and variable density issues.

Contribution

It introduces a novel approach using Lagrangian coordinates, complex interpolation, and maximal regularity in Lebesgue spaces to handle free boundary problems with variable density.

Findings

Established well-posedness for small initial data in critical spaces.

Developed new boundary control techniques using complex interpolation.

Analyzed stability of equilibrium configurations, especially in 2D cases.

Abstract

We study free boundary problems for incompressible inhomogeneous flows governed by the Navier--Stokes equations, focusing on the regularity and global-in-time well-posedness of solutions in critical functional frameworks for small initial data. We introduce a novel analytical framework for free boundary problems formulated as perturbations of the half-space. Our approach relies on the natural Lagrangian change of coordinates and a detailed analysis of the linearized problem (the Stokes system) in the maximal regularity regime, formulated in the Lebesgue spaces $L_p(0,T; L_q)$, including time-weighted variants. The main difficulty lies in the treatment of boundary terms, for which we apply a new technique based on complex interpolation to control nonlinear terms in fractional Sobolev spaces. This strategy also allows us to handle the case of variable density, which is not easily addressed by approaches based on Besov spaces. Using this framework and real interpolation techniques, we construct also solutions in the Lorentz class $L_{p,1}(0,T; L_q)$ in time. The method further enables a rigorous study of the stability of equilibrium configurations. In particular, we resolve the problem in two spatial dimensions, where the interplay between geometry and regularity is especially subtle. Beyond these specific applications, the proposed approach provides a powerful tool for broader classes of nonlinear PDEs and further developments in maximal regularity theory.