The existence of solutions of 2-dimensional incompressible Navier-Stokes equations with surface tension in an optimal Sobolev space

TL;DR

This paper proves the existence of solutions to 2D incompressible Navier-Stokes equations with surface tension on moving domains within an optimal Sobolev space, without requiring compatibility conditions.

Contribution

It establishes the existence of solutions in an optimal Sobolev space for 2D Navier-Stokes with surface tension on moving domains, removing the need for compatibility conditions.

Findings

Existence of solutions in optimal Sobolev space

No compatibility conditions needed

Applicable to moving domains with surface tension

Abstract

We establish the existence of a solution to the Navier-Stokes equations on a moving domain with surface tension in an optimal Sobolev space for the case of two space dimension. No compatibility conditions are required to guarantee the existence of a solution.