Maximal $L^1$-regularity and free boundary problems for the incompressible Navier-Stokes equations in critical spaces

TL;DR

This paper establishes global well-posedness for free boundary incompressible Navier-Stokes equations in critical Besov spaces using maximal L^1-regularity techniques, advancing understanding of viscous fluid dynamics near free surfaces.

Contribution

It introduces a novel approach leveraging maximal L^1-regularity and the structure of the quasi-linear term to prove well-posedness in critical spaces for free boundary problems.

Findings

Global well-posedness for small initial data in critical Besov spaces.

Utilization of maximal L^1-regularity for the Stokes problem in half-space.

Identification of special structures in the quasi-linear term from Lagrangian transform.

Abstract

Time-dependent free surface problem for the incompressible Navier-Stokes equations which describes the motion of viscous incompressible fluid nearly half-space are considered. We obtain global well-posedness of the problem for a small initial data in scale invariant critical Besov spaces. Our proof is based on maximal $L^1$-regularity of the corresponding Stokes problem in the half-space and special structures of the quasi-linear term appearing from the Lagrangian transform of the coordinate.