Clustered vortex helices with compactly supported cross-sectional vorticity in the 3D Euler equations

1. Problem Statement

The paper addresses the rigorous construction of smooth, global-in-time solutions to the three-dimensional incompressible Euler equations in the whole space $\mathbb{R}^3$. Specifically, the authors focus on helical flows without swirl.

The core objective is to construct a solution where the vorticity concentrates around a cluster of collapsing helical filaments as a small parameter $\varepsilon \to 0$. Unlike previous constructions where vorticity cores decay rapidly (exponentially), this work aims to construct solutions where the cross-sectional vorticity remains compactly supported in $\mathbb{R}^2$ for all time.

The dynamics of such filaments are governed asymptotically by the binormal flow:
$$\partial_\tau \gamma = \bar{c} \kappa B_G(\tau)$$
where $\kappa$ is curvature and $B_G$ is the binormal vector. The challenge lies in handling the strong interaction between multiple filaments that are asymptotically close to each other (separation scale $O(|\log \varepsilon|^{-1})$) while maintaining compact support.

2. Methodology

The authors employ a sophisticated Inner-Outer gluing scheme, a perturbative method used to construct exact solutions from approximate ones. The methodology involves several key steps:

A. Reduction to 2D via Helical Symmetry

The 3D Euler equations are reduced to a transport-elliptic system in $\mathbb{R}^2$ by assuming helical symmetry with no swirl. The problem transforms into finding a stream function $\Psi$ satisfying a semilinear elliptic equation:
$$\nabla \cdot (K \nabla \Psi) + F\left(\Psi - \frac{\alpha}{2} |\log \varepsilon| |x|^2\right) = 0 \quad \text{in } \mathbb{R}^2$$
where $K$ is a specific coefficient matrix depending on the helical pitch $h$, and $F$ is a nonlinearity to be chosen.

B. Construction of the Approximate Solution

1. Local Approximation: Near each vortex center $P_i$, the authors construct a local profile using a rescaled solution $\Gamma$ to the semilinear equation $\Delta \Gamma + \Gamma_+^\gamma = 0$ in $\mathbb{R}^2$ (with $\gamma > 3$). This profile naturally possesses compact support.
2. Global Approximation: They superimpose $N$ such local profiles, centered at points $P_i$ that cluster near a limiting circle/helix. A global correction term $H_{2\varepsilon}$ is added to handle the bounded error terms arising from the interaction between vortices and the non-uniform ellipticity of the operator $K$ in the whole space.
3. Scaling Parameters: The scaling parameters $\mu_i$ (controlling the core size) are carefully chosen to eliminate leading-order logarithmic terms in the expansion of the stream function near the vortex centers.

C. The Gluing Scheme (Inner-Outer Decomposition)

The exact solution is sought in the form $\Psi = \Psi_0 + \phi$, where $\Psi_0$ is the approximation and $\phi$ is a perturbation.

• Inner Problem: Solved in rescaled variables near each vortex $P_i$. This involves solving a linearized equation involving the operator $\Delta + \gamma \Gamma_+^{\gamma-1}$.
• Outer Problem: Solved in the whole space $\mathbb{R}^2$ away from the vortices.
• Coupling: The two problems are coupled. A major technical difficulty arises because the linearized operator in the Inner problem has a radial kernel element $Z_0 = O(\log(2+|y|))$ with a non-zero projection $\int \Gamma_+^{\gamma-1} Z_0 \neq 0$. This creates a strong coupling between the Inner and Outer problems, unlike previous works (e.g., [23]) where the kernel was orthogonal to the source term.

D. Resolution of Coupling and Balancing

To resolve the strong coupling:

1. Perturbation of Scaling: The authors introduce small perturbations $\mu_i^*$ to the scaling parameters. These are chosen a posteriori to weaken the coupling and ensure solvability of the projected linear problem.
2. Vortex Location Adjustment: The positions of the vortices $P_i$ are adjusted as small perturbations of a "balancing configuration" $\hat{P}_i$. This configuration must satisfy a system of algebraic equations (balancing conditions) derived from the orthogonality conditions of the linearized operator.
3. Non-degeneracy: The existence of a solution relies on the non-degeneracy of the balancing configuration (the linearization of the balancing system has a one-dimensional kernel due to rotational symmetry).

3. Key Contributions

1. First Smooth Multi-Vortex Helical Solution with Compact Support: This is the first construction of a smooth, global-in-time solution in $\mathbb{R}^3$ where the vorticity concentrates on collapsing helices and the cross-sectional vorticity is compactly supported (as opposed to exponentially decaying).
2. Handling Strong Coupling in Whole Space: The authors overcome the difficulty of constructing solutions in the whole space $\mathbb{R}^3$ (which reduces to $\mathbb{R}^2$) where the Green's function expansion lacks uniform ellipticity. They adapt the gluing technique to handle the non-orthogonal radial kernel $Z_0$ by refining the scaling parameters.
3. Generalization of Collapsing Configurations: The result generalizes previous collapsing vortex configurations (which had rapidly decaying cores) to the case of compact support, allowing for a more physically realistic model of vortex tubes with sharp boundaries.
4. New Nonlinearity Choice: The choice of the nonlinearity $F(s) \sim s_+^\gamma$ (with $\gamma > 3$) is crucial. It differs from the exponential nonlinearity ($e^s$) used in previous works (like [23]), which yielded rapidly decaying vorticity. This choice is what enables the compact support property.

4. Main Results

Theorem 1: For any integer $N \ge 1$, constants $r_0, h > 0$, and circulations $\kappa_i$ with $\sum \kappa_i > 0$, there exists a smooth global-in-time solution $\vec{\omega}_\varepsilon(x, \tau)$ to the 3D Euler equations such that:

• As $\varepsilon \to 0$, the vorticity concentrates near $N$ collapsing helical filaments $G_i(\tau)$.
• The filaments undergo coupled rotation and translation at nearly identical speeds.
• The cross-sectional vorticity is compactly supported in $\mathbb{R}^2$ for all time. Specifically, for any time $\tau$, the support is contained within $N$ disjoint compact balls of radius $O(\varepsilon |\log \varepsilon|^2)$.
• The solution converges in the sense of measures to a sum of Dirac measures supported on the helical curves:
  $$\vec{\omega}_\varepsilon(x, |\log \varepsilon|^{-1}\tau) \rightharpoonup \sum_{i=1}^N \kappa_i \delta_{G_i(\tau)} T_{G_i(\tau)}$$

5. Significance

• Mathematical Rigor: The paper provides a rigorous verification of the vortex filament conjecture for a specific, highly non-trivial class of solutions (collapsing helices with compact support) in the whole space.
• Physical Relevance: Vortex filaments with compact support are often more representative of physical vortex tubes (like smoke rings or tornadoes) than those with infinite tails. This construction bridges the gap between idealized mathematical models and physical intuition.
• Technical Advancement: The development of the Inner-Outer gluing scheme to handle the specific kernel properties of the operator $\Delta + \gamma \Gamma_+^{\gamma-1}$ in the whole space opens new avenues for constructing multi-vortex solutions in unbounded domains where standard variational methods or previous gluing techniques fail due to lack of compactness or uniform ellipticity.
• Extension of Literature: It extends the work on helical vortices from bounded domains or single-filament settings to multi-vortex, strongly interacting, collapsing configurations in $\mathbb{R}^3$.