Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

TL;DR

This paper classifies all smooth solutions to the steady 2D Navier-Stokes equations with separated variables in cone-like domains, revealing solutions with boundary regularity and polynomial solutions in the entire plane.

Contribution

It provides a complete classification of separated variable solutions for the steady 2D Navier-Stokes equations in cone-like domains, including boundary behavior and polynomial solutions.

Findings

Some solutions are Hölder continuous on the boundary but have gradient blow-up at corners.

All solutions in the entire plane are polynomial expressions.

Explicit solutions are obtained with separated variables.

Abstract

We investigate the problem of classification of solutions for the steady Navier-Stokes equations in any cone-like domains. In the form of separated variables, $$u(x,y)=\left( \begin{array}{c} \varphi_1(r)v_1(\theta) \varphi_2(r)v_2(\theta) \end{array} \right) ,$$ where $x=r\cos\theta$ and $y=r\sin\theta$ in polar coordinates, we obtain the expressions of all smooth solutions with $C^0$ Dirichlet boundary condition. In particular, it shows that (i) some solutions are found, which are H\"{o}lder continuous on the boundary, but their gradients blow up at the corner; (ii) all solutions in the entire plane of $\mathbb{R}^2$ like harmonic functions or Stokes equations, are polynomial expressions.