[Paper Review] On Nonhomogeneous Boundary Value Problems for the Stationary Navier--Stokes Equations in 2D Symmetric Semi-Infinite Outlets

Michel Chipot, K. Kaulakyt+2 more|arXiv (Cornell University)|May 27, 2015

Advanced Mathematical Modeling in Engineering19 references3 citations

TL;DR

This paper establishes the existence of at least one weak symmetric solution to the stationary 2D Navier-Stokes equations in symmetric semi-infinite outlets (paraboloidal or channel-like) with nonhomogeneous Dirichlet boundary conditions, even when fluxes over inner and outer boundaries are arbitrarily large, provided only that the total flux is zero. The solution may have finite or infinite Dirichlet integral depending on domain geometry, resolving a long-standing case of Leray's problem in unbounded 2D domains under symmetry.

ABSTRACT

We study the stationary nonhomogeneous Navier--Stokes problem in a two dimensional symmetric domain with a semi-infinite outlet (for instance, either parabo-\\loidal or channel-like). Under the symmetry assumptions on the domain, boundary value and external force we prove the existence of at least one weak symmetric solution without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value ${\bf a}$ over the inner and the outer boundaries may be arbitrarily large. Only the necessary compatibility condition (the total flux is equal to zero) has to be satisfied. Moreover, the Dirichlet integral of the solution can be finite or infinite depending on the geometry of the domain.

Motivation & Objective

• To address the long-open Leray problem for nonhomogeneous boundary value problems in unbounded 2D domains with non-zero fluxes.
• To establish existence of weak solutions without smallness assumptions on fluxes, under symmetry and compatibility conditions.
• To analyze the behavior of the Dirichlet integral (energy) of the solution, showing it may be finite or infinite depending on domain geometry.
• To extend previous results in unbounded domains by allowing arbitrarily large fluxes while maintaining solvability.

Proposed method

• Adopt a symmetric domain with one semi-infinite outlet (paraboloidal or channel-like), ensuring structural invariance under reflection.
• Use a special extension of the boundary data a to satisfy Leray–Hopf type inequalities and enable energy estimates.
• Construct a sequence of approximate solutions in bounded subdomains Ωk, using a Galerkin-type approach with finite-dimensional subspaces.
• Apply compactness arguments via Cantor diagonalization to extract a weakly convergent subsequence in W1,2 spaces.
• Pass to the limit in the weak formulation using test functions with compact support, proving the limit satisfies the integral identity.
• Leverage symmetry and the compatibility condition (total flux = 0) to control growth and ensure convergence without flux smallness.

Experimental results

Research questions

• RQ1 Can weak solutions exist for the stationary 2D Navier-Stokes equations in symmetric unbounded domains with nonhomogeneous boundary data and arbitrarily large fluxes?
• RQ2 Does the total flux being zero suffice for existence of a weak solution in such domains, without requiring flux smallness or zero flux on all components?
• RQ3 How does the geometry of the outlet (paraboloidal vs. channel-like) affect the integrability of the solution’s gradient (i.e., Dirichlet integral)?
• RQ4 Can symmetry of the domain, boundary data, and external force ensure existence even when fluxes are large, without additional smallness assumptions?

Key findings

• A weak symmetric solution exists for the stationary 2D Navier-Stokes equations in symmetric semi-infinite outlets with nonhomogeneous Dirichlet boundary conditions, even when fluxes over inner and outer boundaries are arbitrarily large.
• The only necessary condition is the total flux being zero, confirming that the general outflow condition suffices for solvability.
• The Dirichlet integral of the solution may be finite or infinite depending on the geometric properties of the outlet, particularly the decay rate of the cross-sectional width.
• The solution is constructed via a limiting process on bounded subdomains, using compactness and weak convergence, with convergence established via Cantor diagonalization.
• The method allows for arbitrary flux magnitudes on all boundary components, extending prior results that required flux smallness or zero flux.
• The result generalizes earlier work by Fujita and Morimoto, who required zero boundary data on the outer boundary and Poiseuille-type decay, by removing such restrictions.