On the uniqueness of solution to the steady Euler equations with perturbations

TL;DR

This paper investigates the uniqueness of solutions to perturbed steady Euler equations in multi-dimensional space, demonstrating that under certain decay conditions, the only solution is trivial.

Contribution

It establishes uniqueness results for a broad class of steady Euler equations with perturbations, including special cases like self-similar and Navier-Stokes equations.

Findings

Solutions are unique and trivial under decay conditions.

Applicable to Euler equations with nonlinear terms and Navier-Stokes.

Decay assumptions in $L^q$ spaces imply zero solution.

Abstract

In this paper we study the uniqueness property of solutions to the steady incompressible Euler equations with perturbations in $\Bbb R^N$. Our perturbations include as special cases the Euler equations with a `single signed' nonlinear term, the self-similar Euler equations, and the steady Navier-Stokes equations. For these equations show that suitable decay assumptions at infinity on the solution or its derivatives, imposed by the $L^q$ conditions imply that the only possible solution is zero.