[Paper Review] The existence theorem for steady Navier--Stokes equations in the axially symmetric case

Mikhail V. Korobkov, Konstantin Pileckas+1 more|arXiv (Cornell University)|Oct 28, 2011

Advanced Mathematical Modeling in Engineering8 references4 citations

TL;DR

This paper establishes the existence of weak axially symmetric solutions to the steady Navier-Stokes equations in three-dimensional bounded domains with multiply connected boundaries, under relaxed flux conditions. It proves that solutions exist when the total flux is zero and either all boundary components intersect the axis of symmetry or small-flux components are sufficiently close to the axis, extending prior results that required zero flux on each boundary component.

ABSTRACT

We study the nonhomogeneous boundary value problem for Navier-Stokes equations of steady motion of a viscous incompressible fluid in a three-dimensional bounded multiply connected domain. We prove that this problem has a solution in some axially symmetric cases, in particular, when all components of the boundary intersect the axis of symmetry.

Motivation & Objective

• To address the long-standing open problem of existence of weak solutions to the steady Navier-Stokes equations in 3D bounded domains with multiply connected boundaries under non-zero but total-zero flux conditions.
• To extend previous existence results that required zero flux on each boundary component, which excludes physically relevant configurations with sinks and sources.
• To establish existence theorems under axial symmetry assumptions, particularly when all boundary components intersect the axis of symmetry or when only a subset has small fluxes.
• To provide a rigorous framework for analyzing axially symmetric viscous incompressible flows in domains with complex geometry and nontrivial boundary conditions.

Proposed method

• Utilizes variational methods and weak formulations in Sobolev spaces $W^{1,2}(Ω)$ to analyze the stationary Navier-Stokes system.
• Imposes axial symmetry on both the domain $\Omega$ and the boundary data $\mathbf{a}$, reducing the 3D problem to a system invariant under rotation about the $x_3$-axis.
• Applies the compatibility condition $\sum_{j=0}^N \mathcal{F}_j = 0$ where $\mathcal{F}_j = \int_{\Gamma_j} \mathbf{a} \cdot \mathbf{n} \, dS$, allowing non-zero individual fluxes as long as the total is zero.
• Employs compactness arguments and a priori estimates in weighted Sobolev spaces to handle the nonlinear convective term $(\mathbf{u} \cdot \nabla)\mathbf{u}$.
• Introduces a topological argument involving level sets and connected components of sublevel sets to control the behavior of the velocity field near the axis of symmetry.
• Relies on the theory of Stokes systems and the Leray-Schauder fixed-point theorem in symmetric function spaces to prove existence under the stated flux conditions.

Experimental results

Research questions

• RQ1 Under what conditions on the boundary fluxes does a weak axially symmetric solution to the steady Navier-Stokes equations exist in a 3D bounded domain with multiply connected boundary?
• RQ2 Can the existence of a solution be established when individual boundary components have non-zero flux, provided the total flux is zero?
• RQ3 What role does axial symmetry play in relaxing the classical zero-flux condition on each boundary component?
• RQ4 How do small fluxes on non-axially intersecting boundary components affect the solvability of the Navier-Stokes system?
• RQ5 Can the solution be constructed in domains where all boundary components intersect the axis of symmetry, even with large individual fluxes?

Key findings

• The paper proves the existence of at least one weak axially symmetric solution $\mathbf{u} \in W^{1,2}(\Omega)$ to the steady Navier-Stokes equations under axial symmetry and total flux zero.
• Solutions exist when all boundary components $\Gamma_j$ intersect the axis of symmetry, even if individual fluxes $\mathcal{F}_j$ are arbitrarily large.
• Solutions also exist when only $M$ components intersect the axis and the remaining $N-M$ components have fluxes $|\mathcal{F}_j|$ smaller than a small threshold $\delta = \delta(\nu, \Omega)$ depending on viscosity and domain geometry.
• In the case of axial symmetry without rotation ($u_\theta = 0$), the solution inherits this property, ensuring no azimuthal vorticity.
• The result generalizes prior existence theorems that required $\mathcal{F}_j = 0$ for all $j \geq 1$, thus allowing physically realistic configurations with sinks and sources.
• The proof relies on topological and functional-analytic tools, including level set connectivity and compactness in symmetric Sobolev spaces, to control the nonlinear term and boundary behavior.