Existence of global weak solutions for Navier-Stokes equations with large flux

TL;DR

This paper proves the existence of global weak solutions to the Navier-Stokes equations with large flux in a cylindrical domain, using energy estimates and weighted Sobolev space techniques, advancing understanding of fluid flow with inflow and outflow.

Contribution

It establishes the first proof of global weak solutions for Navier-Stokes with large flux and non-vanishing inflow/outflow using novel energy estimate methods.

Findings

Proved global existence of weak solutions under slip boundary conditions.

Derived energy estimates using the Hopf function.

First step towards global regular solutions with inflow and outflow.

Abstract

Global existence of weak solutions to the Navier-Stokes equation in a cylindrical domain under the slip boundary conditions and with inflow and outflow was proved. To prove the energy estimate, crucial for the proof, we use the Hopf function. This makes us possible to derive such estimate that the inflow and outflow must not vanish as t converges to infinity. The proof requires estimates in weighted Sobolev spaces for solutions to the Poisson equation. Finally, the paper is the first step to prove the existence of global regular special solutions to the Navier-Stokes equations with inflow and outflow.