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Splitting and Merging of Stagnation Points of Solutions to the 2D Navier-Stokes Equations

Isidro Benaroya, Alberto Enciso, Daniel Peralta-Salas

TL;DR

The paper shows that any prescribed finite collection of stagnation-point merging and splitting arcs in $\mathbb{R}^2\times\mathbb{R}_+$ can be realized by a solution to the 2D Navier–Stokes equations up to a small spacetime diffeomorphism, by encoding the pattern as isolated critical points of a stream function $\psi$. The construction proceeds by linearizing to the heat equation, solving a robust local heat problem via Cauchy–Kovalevskaya and parabolic approximation, and then lifting the pattern to the nonlinear NS equation through a small scaling and isotopy, ensuring the correct point types and structural stability. The arguments are extended to the torus $\mathbb{T}^2$, using a scaling to manage compactness. The approach blends analytic approximation, geometric control of critical sets, and stability under perturbations to realize arbitrary admissible stagnation trajectories.

Abstract

We construct solutions to the Navier-Stokes equations on $\mathbf{R}^2$ with an arbitrary number of stagnation points which merge and split along trajectories that can be prescribed freely, up to a small deformation.

Splitting and Merging of Stagnation Points of Solutions to the 2D Navier-Stokes Equations

TL;DR

The paper shows that any prescribed finite collection of stagnation-point merging and splitting arcs in can be realized by a solution to the 2D Navier–Stokes equations up to a small spacetime diffeomorphism, by encoding the pattern as isolated critical points of a stream function . The construction proceeds by linearizing to the heat equation, solving a robust local heat problem via Cauchy–Kovalevskaya and parabolic approximation, and then lifting the pattern to the nonlinear NS equation through a small scaling and isotopy, ensuring the correct point types and structural stability. The arguments are extended to the torus , using a scaling to manage compactness. The approach blends analytic approximation, geometric control of critical sets, and stability under perturbations to realize arbitrary admissible stagnation trajectories.

Abstract

We construct solutions to the Navier-Stokes equations on with an arbitrary number of stagnation points which merge and split along trajectories that can be prescribed freely, up to a small deformation.

Paper Structure

This paper contains 14 sections, 8 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of the main theorem: Links realized as sets of stagnation points
  3. Proof of Theorem \ref{['thm:maintheoremR2']}
  4. Proof of Theorem \ref{['thm:mergingsplittingR2']}.
  5. Proof of Proposition \ref{['prop:heat']}
  6. Part I: Construction of a Noncharacteristic Boundary Surface
  7. Part II: Choice of Adapted Cauchy Data
  8. Step 1: System of equations imposed by the gradient nullity condition (A)
  9. Step 2: Choice of a general ansatz for $f$ and $g$.
  10. Step 3: Final adjustments to the Cauchy data
  11. Proof of Proposition \ref{['prop:approx']}
  12. The Case of the Navier-Stokes Equations on the Torus
  13. Proof of Lemma \ref{['lemma:existence']}
  14. Proof of Lemma \ref{['lemma:FGbeta']}

Key Result

Theorem 1.1

Let us fix a positive integer $r\geq 1$ and some small $\varepsilon>0$. Consider $k$ connecting arcs $\Gamma_i$, which we can assume to be merging for $1\leq i\leq k'$ and splitting for $k'<i\leq k$. Then there exists some initial datum $\psi_0\in C^\infty_c(\mathbb{R}^2)$ for which the associated s

Figures (3)

  • Figure 1: Merging of stagnation points: at time $t_1$, the fluid exhibits two stagnation points (in red and blue), which merge into one (in green) at a later time $t_c$, and subsequently disappear. Splitting corresponds to reflecting the curve across a horizontal plane.
  • Figure 2: Red stagnation points represent eddies (nodes) and blue ones represent saddles; on the left a saddle and a node merge into a degenerate stagnation point that immediately disappears, while on the right a degenerate stagnation point is created and then splits into a saddle and a node.
  • Figure 3: A pair of compact curves realized by stagnation points. The arrows indicate the orientation of the curves.

Theorems & Definitions (14)

  • Definition
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • Proposition 2.3
  • Proposition 2.4
  • Remark 3.1
  • Lemma 3.2
  • ...and 4 more