# The Lagrange multiplier and the stationary Stokes equations

## TL;DR

This paper explains how the pressure in stationary Stokes equations acts as a Lagrange multiplier for the incompressibility constraint in a Hilbert space framework.

## Contribution

It clarifies the theoretical relationship between pressure and the divergence-free condition using the concept of Lagrange multipliers.

## Key findings

- Pressure is the Lagrange multiplier for divergence-free constraint.
- Provides a Hilbert space proof of the relationship.
- Clarifies the mathematical foundation of the Stokes equations.

## Abstract

We briefly discuss the notion of the Lagrange multiplier for a linear constraint in the Hilbert space setting, and we prove that the pressure $p$ appearing in the stationary Stokes equations is the Lagrange multiplier of the constraint $\mathrm{div}\, u =0$.