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Asymmetric Self-similar Spiral Solutions of 2-D Incomressible Euler Equations

This paper constructs nonradial, self-similar solutions to the two-dimensional incompressible Euler equations without assuming rotational symmetry, thereby extending previous m-fold symmetric results and providing analytical counterparts to numerical simulations relevant to the non-uniqueness of solutions.

Original authors: Hyungjun Choi

Published 2026-07-23

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Hyungjun Choi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean of fluid, where everything from the air we breathe to the water in our oceans flows according to strict, unbreakable rules. Scientists call these rules the "Euler equations," and they are the instruction manual for how fluids move when they don't get sticky or squishy. For a long time, mathematicians have been trying to solve a specific puzzle: if you start with a perfectly smooth, calm fluid, can it suddenly twist itself into a chaotic, spiraling mess? Or, if you see a spiral, can you be sure it came from only one specific starting point? This question is crucial because if the answer is "no"—if the same starting point can lead to two different futures—it would mean our current understanding of physics has a blind spot. It's like rolling a ball down a hill and finding out it could end up in two different valleys at the exact same time, which would break the laws of cause and effect.

In this paper, the author, Hyungjun Choi, tackles a very specific type of fluid motion: a self-similar spiral. Think of a self-similar spiral like a perfect, mathematical nautilus shell or a galaxy that looks exactly the same whether you zoom in close or zoom out far; the shape doesn't change, it just scales up or down. Previous scientists had found these spirals, but only if the fluid was perfectly symmetrical, like a star with identical points all around. Choi's work is a major step forward because he proves that these spirals can exist even when the fluid is asymmetrical—when it's lopsided, uneven, and doesn't follow a perfect star pattern. He shows that you can start with a messy, uneven swirl of fluid, and it will still settle into a stable, spiraling shape that follows the rules of the Euler equations. This doesn't mean the universe is chaotic, but it does show that the "instruction manual" allows for much more variety and messiness than we previously thought possible, especially in the context of whether a fluid's future is truly unique or if it could branch off in different directions.

The Story of the Lopsided Spiral

To understand what Choi did, we first need to meet the main characters: the Euler equations and the vorticity. Imagine the Euler equations as the traffic laws for a fluid. They tell the fluid how to move, how to speed up, and how to turn. "Vorticity" is just a fancy word for how much the fluid is spinning at any given point. If you stir your coffee, the swirl you see is vorticity.

For decades, mathematicians have been hunting for "self-similar solutions." These are special, rare flows where the shape of the swirl stays the same over time, even as the whole thing grows or shrinks. It's like watching a video of a spinning top that speeds up, but if you slow the video down, it looks exactly like the original top, just bigger. The paper focuses on a specific kind of these spirals called "algebraic spirals." These aren't the smooth, logarithmic spirals you see in seashells; they are sharper, more angular, and they emerge instantly from a starting point, almost like a vortex sheet "rolling up" into a spiral.

The big question Choi addresses is about symmetry. Before this paper, we knew these spirals existed, but only if the starting fluid was perfectly symmetrical (like a perfect snowflake with $N$ identical arms). The math was much easier if you assumed the fluid looked the same no matter how you rotated it. But real fluids aren't always perfect snowflakes. They can be lopsided. The big mystery was: Can these spirals form if the starting fluid is messy and asymmetrical?

Choi's main finding is a resounding yes. He constructed a mathematical proof showing that these self-similar spiral solutions exist even when the initial fluid has no rotational symmetry at all. He didn't just guess; he built a rigorous mathematical framework to prove that if you start with a specific type of uneven swirl (mathematically described as a function in a space called $L^1$ with a small amount of "noise" or irregularity), the fluid will evolve into a stable, non-radial spiral.

How the Magic Happens

To pull this off, Choi had to get creative with his tools. He used a technique called "adapted coordinates." Imagine trying to describe the path of a rollercoaster that is twisting and turning wildly. If you try to describe it using a standard map (like latitude and longitude), the lines get messy and hard to follow. Instead, Choi invented a new, custom map that moves with the fluid. In this new coordinate system, the twisting, turning path of the fluid looks like a straight, calm line. This made the complex, twisting math much easier to handle.

He then took a known, simple solution (a perfectly symmetrical spiral) and asked, "What happens if I wiggle it just a tiny bit?" He treated the wiggles as a small disturbance. Using a method called the "implicit function theorem" (which is basically a fancy way of saying "if a small change in input leads to a small, predictable change in output, we can find a solution"), he proved that even with these wiggles, the system doesn't fall apart. The spiral holds its shape.

Crucially, he didn't just assume the wiggles were small; he proved that as long as the "messiness" of the starting fluid is within a certain limit (mathematically, the sum of the Fourier coefficients weighted by $|n|^{-1/2}$ is small enough), the spiral will form. This is a significant improvement over previous work, which required the fluid to have a high degree of symmetry (like having 100 or 1,000 identical arms) to make the math work. Choi removed that requirement entirely.

Why This Matters (and What It Doesn't)

So, why should a curious teenager care? This paper is a piece of a much larger puzzle regarding non-uniqueness. In the world of fluid dynamics, there is a lingering fear that the Euler equations might not always give a single, unique answer. If you start with a specific fluid state, could it evolve into two completely different futures? This would be a disaster for physics, as it would mean the future is not determined by the present.

Choi's work supports the idea that these spirals are real and stable, even in messy conditions. This is relevant to numerical simulations (computer models) done by other scientists, like Bressan and Shen, who saw that a single starting point might split into two different spiral patterns. Choi's paper suggests that the "single-wing" spiral is a mathematically valid possibility, even without perfect symmetry. It doesn't prove that the universe is non-unique (that's still an open question), but it proves that the "rules" allow for these complex, asymmetric shapes to exist.

It's important to note what this paper doesn't do. It doesn't simulate a real fluid in a lab or on a computer; it's a pure mathematical proof. It doesn't say that every messy fluid will turn into a spiral, only that some specific types of messy fluids will. And it doesn't solve the mystery of non-uniqueness once and for all; it just adds a new, robust piece of evidence to the table.

In the end, Choi has shown that the universe of fluid dynamics is more flexible than we thought. You don't need a perfect, symmetrical snowflake to create a beautiful, mathematical spiral. A little bit of asymmetry, a little bit of mess, and the laws of physics can still spin a perfect, self-similar dance. This opens the door for future mathematicians to explore even more chaotic and realistic fluid behaviors, bringing us one step closer to understanding the true, messy beauty of the flowing world around us.

Technical Summary: Asymmetric Self-similar Spiral Solutions of 2-D Incompressible Euler Equations

Problem Statement
This paper addresses the existence of non-radial, self-similar solutions to the two-dimensional incompressible Euler equations. The study focuses on solutions of the form $\omega(t, x) = t^{-1}\Omega(t^{-\mu}x)$ and $u(t, x) = t^{-1+\mu}U(t^{-\mu}x)$ , where the vorticity profile $\Omega$ is singular at the origin. While previous works by Elling [23, 24] and Shao–Wei–Zhang [41] established the existence of such algebraic spiral solutions, they relied on the assumption of $m$ -fold rotational symmetry (with $m \ge 2$ ). The central problem tackled here is the construction of these solutions without assuming any rotational symmetry, thereby addressing a more general class of initial data. This work is motivated by the ongoing investigation into the non-uniqueness of weak solutions in 2-D fluid dynamics, particularly in connection with numerical simulations by Bressan and Shen [10] which suggest that a single initial datum might evolve into distinct self-similar spiral structures.

Methodology
The authors employ a perturbation analysis around a known radial self-similar solution. The methodology proceeds through the following steps:

Adapted Coordinates: Following the approach of Elling, the authors introduce a change of coordinates $(r, \theta) \to (\beta, \phi)$ , where $\phi$ is constant along the characteristic curves of the vector field $\nabla^\perp \Psi - \mu x$ . This transformation reduces the self-similar Euler equations to a nonlinear elliptic equation for a stream function profile $f$ in the new coordinates.
Linearization and Functional Spaces: The core of the analysis involves the linearization of the resulting nonlinear operator $F(\Gamma, f) = 0$ around the trivial radial solution $(\Gamma_0, f_0)$ . A critical innovation is the choice of function spaces. Unlike previous works that utilized Hölder spaces or relied on symmetry to handle low-frequency modes, this paper defines Wiener-type spaces ( $X, Y, Z$ ) based on the $A^\alpha$ norms (weighted sums of Fourier coefficients). These spaces are specifically designed to handle the lack of rotational symmetry.
Invertibility of the Linearized Operator: The authors prove that the linearized operator $L: Y \to Z$ $L : Y \to Z$ is an isomorphism for the parameter regime $\mu > 1$ $μ > 1$ . This is achieved by:
- Decomposing the operator into a "main part" and a "perturbation."
- Showing that the perturbation is relatively compact with respect to the main part, a property that fails in previous decompositions for small Fourier modes.
- Utilizing the Fredholm alternative to establish invertibility for all Fourier modes, including the critical $|n|=1$ modes which were problematic in earlier studies for $\mu > 1/2$ .
- Proving uniqueness (injectivity) of the solution to the linearized problem, which relies on the condition $\mu > 1$ to ensure the triviality of the kernel.
Nonlinear Analysis: Using the Implicit Function Theorem (or a contraction mapping argument) in the constructed Banach spaces, the authors establish the existence of a unique solution $f$ for small perturbations of the radial profile.
Reconstruction: The solution is mapped back to the original physical coordinates. The authors verify that the coordinate transformation is a $C^1$ -diffeomorphism and that the resulting velocity and vorticity fields satisfy the weak formulation of the Euler and vorticity equations, respectively.

Key Contributions and Results

Main Theorem: The paper proves that for any $\mu > 1$ , there exists a sufficiently small $\epsilon > 0$ such that for any initial vorticity profile $\mathring{\omega} \in L^1(\mathbb{T})$ satisfying a smallness condition in the $A^{-1/2}$ norm (specifically $\sum_{n \neq 0} |n|^{-1/2} |\hat{\mathring{\omega}}_n| \le \epsilon |\hat{\mathring{\omega}}_0|$ ), there exists a global weak solution to the 2-D Euler equations.
Removal of Symmetry: Unlike Theorem 1.1 of Elling and Theorem 1.2 of Shao–Wei–Zhang, this result does not require the initial data to possess $N$ -fold rotational symmetry. The smallness condition replaces the symmetry requirement, allowing for fully asymmetric spiral solutions.
Regularity and Convergence: The constructed solutions are continuous in time with values in $L^2_{loc}$ for velocity and $L^1_{loc}$ for vorticity. The initial data is recovered in the limit $t \to 0^+$ in the appropriate distributional and local $L^p$ senses.
Extension to Measures: The result is extended to cases where the initial vorticity is a bounded Radon measure (via mollification), broadening the scope of admissible initial data.

Significance
The paper claims significance primarily in the context of the non-uniqueness of weak solutions to the 2-D incompressible Euler equations. By constructing self-similar solutions without rotational symmetry, the work provides analytical support for the plausibility of the "single-wing" spiral scenario observed in Bressan–Shen numerical simulations [10]. The authors note that these solutions emerge instantaneously from the initial data (a phenomenon analogous to vortex-sheet roll-up), contrasting with logarithmic spirals where the structure exists at $t=0$ .

The work also highlights a technical advancement in the analysis of linearized operators for fluid equations. By utilizing Wiener-type spaces and a specific decomposition that ensures compactness of the perturbation, the authors overcome the difficulties associated with low-frequency modes that previously necessitated symmetry assumptions. This approach allows for the treatment of the regime $\mu > 1$ , which is distinct from the $\mu > 1/2$ regime studied in symmetric settings, and opens a pathway for future investigations into non-uniqueness mechanisms that respect the underlying transport structure of the Euler equations.

The Story of the Lopsided Spiral

How the Magic Happens

Why This Matters (and What It Doesn't)

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