Long time existence for nonhomogeneous Navier–Stokes equations with large flux
Dissertationes Mathematicae (2026)
MSC: Primary 35Q30; Secondary 76D03, 76D05.
DOI: 10.4064/dm240822-23-2
Published online: 4 October 2026
Abstract
The nonhomogeneous Navier–Stokes equations are examined in a cylindrical domain in $\mathbb{R}^3$, parallel to the $x_3$-axis, with large inflow and outflow at the top and bottom of the cylinder. Additionally, slip boundary conditions are applied to the lateral surface of the cylinder. This paper establishes the long time existence of regular solutions under the condition that the inflow and outflow are close to homogeneous, and that the norms of the derivatives with respect to $x_3$ of both the external force and the initial velocity are sufficiently small. A key point of this study is to demonstrate that the $x_3$-component of the velocity remains positive.
