Global solvability of the Navier-Stokes equations with a free surface in the maximal $L_p\text{-}L_q$ regularity class

TL;DR

This paper proves the global solvability of the Navier-Stokes equations with a free surface in a maximal regularity class, using linearized stability and nonlinear analysis.

Contribution

It establishes the global existence and stability of solutions for free surface Navier-Stokes equations in the maximal Lp-Lq regularity framework, without gravity.

Findings

Global solvability in maximal Lp-Lq class

Exponential stability of solutions

Applicability to free surface fluid flows

Abstract

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the $N$-dimensional Euclidean space for $N\geq 2$ when the gravity is not taken into account. The aim of this paper is to show the global solvability of the Naiver-Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}L_q$ regularity class. Our approach is based on the maximal $L_p\text{-}L_q$ regularity with exponential stability for the linearized equations, and solutions to the original nonlinear problem are also exponentially stable.