Initial-boundary value problem of the Navier-Stokes system in the half space

TL;DR

This paper establishes the existence and uniqueness of weak solutions to the Navier-Stokes initial-boundary value problem in a half space, with solutions exhibiting specific regularity properties, extending previous results to nonhomogeneous boundary conditions.

Contribution

It proves unique solvability of the Navier-Stokes system with nonhomogeneous boundary data in a half space, generalizing prior work.

Findings

Unique weak solution exists on short time interval

Solution has Hölder continuity in space and time

Extends previous results to nonhomogeneous boundary conditions

Abstract

In this paper, we study the initial-boundary value problem of the Navier-Stokes system in the half space. We prove the unique solvability of the weak solution on some short time interval (0, T) with the velocity in $C^{\alpha, \frac12 \alpha} ({\mathbb R}^n_+ \times (0, T)), 0 < \alpha < 1$, when the given initial data is in $C^\alpha ({\mathbb R}^n_+)$ and the given boundary data is in $C^{\alpha, \frac12 \alpha} ({\mathbb R}^{n-1} \times (0, T))$. Our result generalizes the result in [30] considering nonhomogeneous Dirichlet boundary data.