Stationary Navier-Stokes Equations, results and problems

February 04, 2025

Stationary Navier-Stokes Equations, results and problems

In this blog, we look into the stationary Navier-Stokes equation:

$$ \begin{cases} -\nu \Delta u + (u\cdot \nabla) u + \nabla p = 0 \\ \text{div } u = 0 \end{cases} $$

where $ u: \mathbb{R}^n\to \mathbb{R}^n $ is a vector field, $p:\mathbb{R}^n \to \mathbb{R}$ is a function. Both $ u $ and $ p $ are unknown.

Stationary Navier-Stokes has been investigated for more than a century and we still know little about it until recently.

Many problems arise and still open in dimension $ n=2,3 $.

Boundary Value Problem

Given $ \Omega \subset \mathbb{R}^n $ be a bounded open domain, with $ C^1 $ boundary. The BVP come out naturally:

$$ \begin{cases} -\nu \Delta u + (u\cdot \nabla) u + \nabla p = 0 \quad \text{in } \Omega \\ \text{div } u = 0 \quad \text{in } \Omega \\ u = a \quad \text{on } \partial\Omega \end{cases} $$

The fundamental work is Leray's theory in 1930s. The most important tool is the Leray-Schauder fixed point theorem:

Let $ H $ be a Hilbert space and $ K:H\to H $ be a compact operator. Then for the equation $ x=Kx $, there exists a solution if for all $ \lambda \in [0,1] $, the possible solutions to $ x_{\lambda} = \lambda Kx_{\lambda} $ are uniformly bounded, i.e.

$$ \sup_{\lambda\in[0,1]} \Vert x_{\lambda} \Vert_H \le +\infty $$

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