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The steady Navier-Stokes equations in a system of unbounded channels with sources and sinks

Filippo Gazzola, Mikhail V. Korobkov, Xiao Ren, Gianmarco Sperone

TL;DR

This work extends Leray–Ladyzhenskaya analysis to a planar, non-simply connected junction of unbounded channels with inhomogeneous boundary data and large fluxes. It develops a flux-carrier construction and a Leray–Hopf-type inequality via a Leray reductio argument, then uses invading-domain techniques to obtain a strong solution with a uniform Dirichlet-energy bound on compact sets. For small data, it proves asymptotic convergence to Couette–Poiseuille flows in each outlet and establishes uniqueness in this regime. The results generalize classical theories to more complex geometries and boundary conditions, providing rigorous existence, asymptotics, and uniqueness in physically relevant channelized flows. The methods combine PDE analysis in unbounded domains, Euler limit analysis, and Morse–Sard-type structural results for Sobolev solutions, with potential applications to flows around bridges and irrigation networks.

Abstract

The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous works, the domain is not assumed to be simply-connected and the fluxes are not assumed to be small. In this very general setting, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. This is a generalization of the classical Ladyzhenskaya-Solonnikov result obtained under the additional assumption of zero boundary conditions. For small data of the problem we also prove the unique solvability and attainability of Couette-Poiseuille flows at infinity. The main novelty of our approach is the proof of the corresponding Leray-Hopf-type inequality by Leray's reductio ad absurdum argument (since the standard Hopf cutoff extension procedure does not work for general boundary data). For this contradiction approach, we use some fine properties of weak solutions to the Euler system based on Morse-Sard-type theorems in Sobolev spaces obtained by Bourgain, Korobkov & Kristensen.

The steady Navier-Stokes equations in a system of unbounded channels with sources and sinks

TL;DR

This work extends Leray–Ladyzhenskaya analysis to a planar, non-simply connected junction of unbounded channels with inhomogeneous boundary data and large fluxes. It develops a flux-carrier construction and a Leray–Hopf-type inequality via a Leray reductio argument, then uses invading-domain techniques to obtain a strong solution with a uniform Dirichlet-energy bound on compact sets. For small data, it proves asymptotic convergence to Couette–Poiseuille flows in each outlet and establishes uniqueness in this regime. The results generalize classical theories to more complex geometries and boundary conditions, providing rigorous existence, asymptotics, and uniqueness in physically relevant channelized flows. The methods combine PDE analysis in unbounded domains, Euler limit analysis, and Morse–Sard-type structural results for Sobolev solutions, with potential applications to flows around bridges and irrigation networks.

Abstract

The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous works, the domain is not assumed to be simply-connected and the fluxes are not assumed to be small. In this very general setting, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. This is a generalization of the classical Ladyzhenskaya-Solonnikov result obtained under the additional assumption of zero boundary conditions. For small data of the problem we also prove the unique solvability and attainability of Couette-Poiseuille flows at infinity. The main novelty of our approach is the proof of the corresponding Leray-Hopf-type inequality by Leray's reductio ad absurdum argument (since the standard Hopf cutoff extension procedure does not work for general boundary data). For this contradiction approach, we use some fine properties of weak solutions to the Euler system based on Morse-Sard-type theorems in Sobolev spaces obtained by Bourgain, Korobkov & Kristensen.

Paper Structure

This paper contains 15 sections, 19 theorems, 217 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['th:main']}
  4. The proof at a glance
  5. Construction of a flux carrier
  6. Leray's method of invading domains
  7. The global linear growth of the Dirichlet integral
  8. The uniform linear growth of the Dirichlet integral
  9. The uniform local estimates of the Dirichlet integral
  10. Conclusion of the proof of Theorem \ref{['th:main']}
  11. Proof of Theorem \ref{['th:main2']}
  12. Proof of Theorem \ref{['th:main3']}
  13. Two physical models
  14. Appendix (the construction of the flux carrier).
  15. Appendix (entire solutions to the Euler equations).

Key Result

Theorem 2.2

Let $\Omega \subset \mathbb{R}^2$ be an admissible domain. Suppose that ${\mathbf f}\in L_{\mathrm{loc}}^{2}(\overline\Omega)$ is an external force verifying uef, ${\mathbf a}\in W^{3/2,2}_{\mathrm{loc}}(\partial\Omega)$ is a boundary velocity obeying bda-bda-1, and ${\mathcal{F}}_1,...,{\mathcal{F} where $C_* > 0$ depends on $\Omega$, ${\mathbf f}$, ${\mathbf a}$ and $\{\mathcal{F}_j\}^{J}_{j=1}$

Figures (9)

  • Figure 1.1: A smooth, unbounded and distorted channel.
  • Figure 1.2: Junction of unbounded channels considered by Ladyzhenskaya & Solonnikov, with $J=4$.
  • Figure 2.1: An admissible domain $\Omega$ with $I=3$ and $J=4$.
  • Figure 2.2: Decomposition of the boundary of an outlet $\Omega_{j}$, for $j \in \{1,\dots,J \}$.
  • Figure 3.1: The truncated domain $\Omega^2$, defined in (\ref{['Omega_ab']}), with $I=3$, $J=4$ and $\Gamma_5 = \Gamma_1$.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 2.2: Existence
  • Remark 2.3
  • Theorem 2.4: Asymptotics
  • Theorem 2.5: Uniqueness
  • Remark 2.6
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • ...and 17 more