Navier-Stokes Equations 1
Recent Advances in Steady States of Navier-Stokes Equations
The study of steady-state and self-similar solutions of the incompressible Navier-Stokes equations (NSE) has undergone remarkable progress in the 2020s. This post surveys landmark results from 2024–2026 touching on existence, uniqueness, classification, and stability of such solutions. The stationary (steady) NSE in $\mathbb{R}^3$ reads: $$-\nu \Delta u + (u \cdot \nabla) u + \nabla p = 0, \quad \operatorname{div} u = 0.$$ A central object of the self-similar theory is the class of $(-1)$-homogeneous (scale-invariant) solutions: a function $u$ is $(-1)$-homogeneous if $u(\lambda x) = \lambda^{-1} u(x)$ for all $\lambda > 0$. These are precisely the profiles of forward self-similar solutions $u(x,t) = t^{-1/2} U(x/\sqrt{t})$ of the time-dependent NSE.