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On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions

Jeaheang Bang, Changfeng Gui, Hao Liu, Yun Wang, Chunjing Xie

TL;DR

This work establishes the existence of self-similar $(-1)$-homogeneous solutions to the steady Navier–Stokes equations in dimensions $4\le n\le 16$ under a $(-3)$-homogeneous, locally Lipschitz external force $\boldsymbol f$, with regularity away from the origin. The authors develop a novel a priori energy framework that links the radial velocity component to the total head pressure on the sphere, enabling control of nonlinear terms and higher integrability of $H_+$, and they close the estimates via a Leray–Schauder degree argument to obtain existence; global uniqueness is shown when the forcing is small, and a dimension-free result is achieved for forces with nonnegative radial component. The approach combines sphere-based identities, decomposition into tangential and radial parts, and elliptic regularity to bootstrap regularity up to $C^{2,\alpha}_{loc}$, extending Shi’s 4D results to higher dimensions. This advances understanding of scale-invariant steady flows under large, translation-invariant forcing and provides a rigorous fixed-point construction for self-similar solutions in higher-dimensional settings.

Abstract

We prove that the steady incompressible Navier-Stokes equations with any given $(-3)$-homogeneous, locally Lipschitz external force on $\mathbb{R}^n\setminus\{0\}$, $4\leq n\leq 16$, have at least one $(-1)$-homogeneous solution which is scale-invariant and regular away from the origin. The global uniqueness of the self-similar solution is obtained as long as the external force is small. The key observation is to exploit a nice relation between the radial component of the velocity and the total head pressure under the self-similarity assumption. It plays an essential role in establishing the energy estimates. If the external force has only the nonnegative radial component, we can prove the existence of $(-1)$-homogeneous solutions for all $n\geq 4$. The regularity of the solution follows from integral estimates of the positive part of the total head pressure, which is due to the maximum principle and a ``dimension-reduction" effect arising from the self-similarity.

On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions

TL;DR

This work establishes the existence of self-similar -homogeneous solutions to the steady Navier–Stokes equations in dimensions under a -homogeneous, locally Lipschitz external force , with regularity away from the origin. The authors develop a novel a priori energy framework that links the radial velocity component to the total head pressure on the sphere, enabling control of nonlinear terms and higher integrability of , and they close the estimates via a Leray–Schauder degree argument to obtain existence; global uniqueness is shown when the forcing is small, and a dimension-free result is achieved for forces with nonnegative radial component. The approach combines sphere-based identities, decomposition into tangential and radial parts, and elliptic regularity to bootstrap regularity up to , extending Shi’s 4D results to higher dimensions. This advances understanding of scale-invariant steady flows under large, translation-invariant forcing and provides a rigorous fixed-point construction for self-similar solutions in higher-dimensional settings.

Abstract

We prove that the steady incompressible Navier-Stokes equations with any given -homogeneous, locally Lipschitz external force on , , have at least one -homogeneous solution which is scale-invariant and regular away from the origin. The global uniqueness of the self-similar solution is obtained as long as the external force is small. The key observation is to exploit a nice relation between the radial component of the velocity and the total head pressure under the self-similarity assumption. It plays an essential role in establishing the energy estimates. If the external force has only the nonnegative radial component, we can prove the existence of -homogeneous solutions for all . The regularity of the solution follows from integral estimates of the positive part of the total head pressure, which is due to the maximum principle and a ``dimension-reduction" effect arising from the self-similarity.

Paper Structure

This paper contains 9 sections, 16 theorems, 132 equations.

Table of Contents

  1. Introduction and main results
  2. Existence and uniqueness of self-similar solutions
  3. Main ideas and outlines for the proof
  4. Organization of the paper
  5. A priori energy estimates
  6. Energy estimates
  7. Proof of the main theorem
  8. The case that the forces have only nonnegative radial components
  9. Two technical issues

Key Result

Theorem 1.1

For $n\in\mathbb{N}$, $4 \leq n \leq 16$, let ${\boldsymbol{f}}$ be a $(-3)$-homogeneous force such that ${\boldsymbol{f}}$ is locally Lipschitz on $\mathbb{R}^n \setminus\{0\}$. Then we have the following results.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 25 more