2D capillary liquid drops with constant vorticity: rotating waves existence and a conditional energetic stability result for rotating circles

This paper establishes the existence and conditional energetic stability of rotating wave solutions for two-dimensional capillary liquid drops with constant vorticity by utilizing a Hamiltonian framework, bifurcation theory, and critical point theory.

The Gist

Imagine a drop of liquid floating in space, like a tiny, perfect sphere of water. Now, imagine that this drop isn't just sitting still; it's spinning, wobbling, and changing shape, all while held together by surface tension (the "skin" that makes water bead up).

This paper is a mathematical detective story about what happens when we add spin (vorticity) to this spinning liquid drop. The author, Giuseppe La Scala, asks three big questions:

1. Can we predict how this drop moves?
2. Can we find special shapes where the drop spins in a perfect, repeating pattern (like a rotating wave)?
3. If we nudge the drop slightly, will it snap back to its original shape, or will it fly apart?

Here is the breakdown of the paper's findings using simple analogies.

1. The Setup: A Spinning Pizza Dough

Think of the liquid drop as a piece of pizza dough.

• The Shape: Usually, we think of a drop as a perfect circle. But in reality, it can stretch and squish. The author describes this shape using a "height map" (how much the dough sticks out from the center).
• The Spin: The drop has two types of motion.
  • The Flow: The water moving around inside (like dough being tossed).
  • The Vorticity (The Twist): The author adds a constant "twist" to the whole system. Imagine the entire pizza dough is being spun on a lazy Susan while the dough itself is being tossed. This "constant vorticity" makes the math much harder because the fluid is rotating internally, not just moving as a whole.

2. The Map: Translating the Chaos

The equations governing this drop are incredibly complex, like trying to write a recipe for a soufflé that changes its own ingredients while baking.

• The Craig-Sulem Equations: The author translates these messy physics equations into a cleaner language called the "Craig-Sulem formulation." Think of this as converting a chaotic, handwritten note into a clear, typed spreadsheet.
• The Flat Torus: To make the math easier, he imagines the drop's surface isn't a circle, but a "flat torus" (like a video game screen where if you go off the right edge, you reappear on the left). This removes the "curvature" headaches and lets him use standard tools.

3. The Energy Engine: The Hamiltonian

The paper discovers that this spinning drop is a Hamiltonian system.

• The Analogy: Imagine a giant, invisible energy machine. The drop has "Kinetic Energy" (from moving) and "Potential Energy" (from surface tension trying to make it round).
• The Twist: Because of the internal spin (vorticity), the author had to invent a new "Energy Score" (Hamiltonian) that includes a penalty for the drop's area. It's like a video game where you get points for speed, but you lose points if your character gets too big.
• Conservation Laws: Just like a spinning top conserves its angular momentum, this drop conserves its Total Angular Momentum and its Volume. These are the "rules of the game" that never change.

4. Finding the "Dancing" Shapes (Rotating Waves)

The author wanted to know: Are there specific shapes where the drop spins perfectly without changing its form?

• The Bifurcation: Imagine a spinning top. At slow speeds, it's a perfect circle. As you speed it up, it might suddenly wobble into a triangle, then a square, then a star. These sudden changes are called "bifurcations."
• The Discovery: The author proved that if you spin the drop at just the right speed (which depends on how strong the surface tension is vs. how strong the internal spin is), the drop can settle into a Rotating Wave.
  • These aren't just circles; they can be shapes with 3-fold symmetry (like a triangle), 4-fold (like a square), etc.
  • He used a mathematical technique called Lyapunov-Schmidt decomposition. Think of this as separating the "main dancer" (the shape of the drop) from the "background noise" (tiny ripples). He showed that if the main dancer hits a specific rhythm, a new, stable dance move appears.

5. The Stability Test: Will it Survive?

This is the most critical part. If you have a perfect rotating drop and you poke it, does it recover?

• The Problem: The author found that for some spins, the drop is actually unstable. It's like a pencil balanced on its tip; a tiny nudge makes it fall. Specifically, if the internal spin is too strong compared to the surface tension, the drop wants to break apart.
• The "Conditional" Fix: However, the author found a way to save the day. He said: "If we promise to keep the drop's Volume and its Center of Mass exactly the same as the perfect circle, then the drop IS stable."
  • The Analogy: Imagine a wobbling spinning top. If you hold the table perfectly still (fixing the center) and make sure no water spills out (fixing the volume), the top might wobble but it won't fall over.
  • This is called Conditional Energetic Stability. The drop is stable only if you don't let it change its size or drift away.

Summary of the Big Picture

• The Physics: A spinning liquid drop with internal rotation is a complex dance between surface tension (trying to make it round) and centrifugal force (trying to fling it apart).
• The Math: The author built a new energy map to track this dance and proved that "perfectly spinning shapes" (rotating waves) exist, even with the internal spin.
• The Result: These shapes are stable, but only if you strictly control their size and position. If you let them drift or change size, the internal spin might tear them apart.

In a nutshell: The paper proves that even a liquid drop with a complicated internal spin can find a stable, rhythmic dance, provided we keep it in a fixed box and don't let it change its size. It connects the physics of spinning fluids to the geometry of shapes, showing that nature loves symmetry, even in chaos.

Technical Summary

Here is a detailed technical summary of the paper "2D capillary liquid drops with constant vorticity: rotating waves existence and a conditional energetic stability result for rotating circles" by Giuseppe La Scala.

1. Problem Statement

The paper investigates the dynamics of two-dimensional, pure capillary liquid drops with constant vorticity ($\alpha_0$). The fluid is modeled as an incompressible, perfect (inviscid) fluid governed by the Euler equations, with a free boundary determined by surface tension (capillarity).

The specific free boundary problem is defined by:

• Governing Equations: Incompressibility ($\nabla \cdot u = 0$) and constant vorticity ($\nabla \times u = \alpha_0$) within the domain $\Omega_t$.
• Boundary Conditions: The pressure on the boundary $\partial \Omega_t$ is determined by the curvature (Young-Laplace law), and the boundary moves with the normal component of the fluid velocity.
• Geometry: The drop is assumed to be a perturbation of a unit circle, described by an elevation function $h(t, x)$ such that the boundary is $\gamma_t(x) = (1+h(t,x))x$.

The primary objectives are:

1. To establish the Hamiltonian structure of the system in the presence of constant vorticity.
2. To prove the existence of rotating wave solutions (non-trivial steady profiles in a rotating frame) bifurcating from the circular state.
3. To analyze the conditional energetic stability of the rotating circular solution.

2. Methodology

A. Mathematical Formulation

The author employs the Craig-Sulem formulation, reducing the free boundary problem to a system of equations on the boundary (the unit circle $S^1$ or the flat torus $T^1$).

• Variables: The system is described by the elevation function $\xi$ (related to $h$) and the trace of the velocity potential $\chi$ (related to $\psi$).
• Transformation: The equations are rewritten on the flat torus $T^1$ using a logarithmic transformation $\xi = \log(1+h)$.
• Hamiltonian Structure: The total energy (Kinetic + Capillary) is identified as the Hamiltonian. However, due to the constant vorticity term, the standard symplectic structure is not directly invertible. The author introduces a change of coordinates (analogous to Wahlén coordinates) to transform the system into a strictly Hamiltonian form on the phase space $H^{s_1} \times \dot{H}^{s_2}$.
  • The new Hamiltonian $\bar{H}$ includes terms for kinetic energy, surface tension, and a penalization term related to the drop's area and vorticity.
  • The Poisson tensor becomes block-diagonal and invertible in the new coordinates.

B. Symmetries and Conserved Quantities

The Hamiltonian structure reveals several symmetries and conserved quantities:

• Translation in Potential: Conservation of Volume ($V$).
• Rotational Invariance: Conservation of Total Angular Momentum ($I$).
• Reflection Symmetry: Invariance under $R(\zeta, \gamma) = (\zeta(-\vartheta), -\gamma(-\vartheta))$.
• Barycenter Velocity: The velocity of the fluid's barycenter is shown to be a conserved quantity, which implies the barycenter position is constant if the velocity is zero.

C. Existence of Rotating Waves

The existence of rotating waves (solutions of the form $\xi(t, \vartheta) = \eta(\vartheta + \omega t)$) is proven using Bifurcation Theory combined with Critical Point Theory.

• Linearization: The system is linearized around the trivial solution (the rotating circle). The eigenvalues are purely imaginary, indicating linear stability.
• Resonance: Bifurcation occurs when the rotation frequency $\omega$ satisfies a specific resonance relation involving the vorticity $\alpha_0$, surface tension $\sigma_0$, and the mode number $\ell$.
• Lyapunov-Schmidt Decomposition: The problem is reduced to a finite-dimensional bifurcation equation on the kernel of the linearized operator.
  • Case 1 (Simple Eigenvalues/Multiplicity 2): The Crandall-Rabinowitz theorem is applied. The transversality condition is verified, proving the existence of a branch of rotating waves.
  • Case 2 (Multiplicity 4): When the kernel dimension is 4, the author uses two approaches:
    • Moser-Weinstein Argument: If $\omega > \alpha_0/2$, the reduced angular momentum is positive-definite, allowing the use of minimax principles to find at least two distinct orbits.
    • Topological Approach: If $\omega < \alpha_0/2$, the author employs Conley index theory and Mountain Pass arguments to establish the existence of critical points even when ellipticity is lost.

D. Stability Analysis

The stability of the rotating circle is analyzed via the Energy-Casimir method (Lyapunov stability).

• The Hessian of the Hamiltonian $\bar{H}$ is computed.
• Instability Directions: The analysis reveals two potential sources of instability:
  1. A zero eigenvalue associated with the volume mode $(\ell, m) = (0,0)$.
  2. A potentially negative eigenvalue associated with the barycenter modes $(\ell, m) = (1, \pm 1)$, depending on the "Modified Bond Number" $C = \sigma_0 / \alpha_0^2$.
• Conditional Stability: The author proves that if the perturbations are constrained to preserve the Volume, Barycenter Velocity, and Barycenter Position, the negative directions are eliminated (becoming higher-order small terms). Under these constraints, the Hamiltonian becomes strictly convex, ensuring conditional energetic stability.

3. Key Contributions

1. Hamiltonian Formulation with Vorticity: The paper successfully derives a rigorous Hamiltonian formulation for 2D capillary drops with constant vorticity, resolving the non-invertibility of the Poisson tensor via a specific coordinate transformation (Wahlén-type).
2. Existence of Rotating Waves: It provides a comprehensive proof of the existence of rotating waves bifurcating from the circular state. It handles both simple eigenvalues and the more complex case of double and quadruple eigenvalues (multiplicity 4), utilizing advanced bifurcation techniques (Crandall-Rabinowitz, Moser-Weinstein, and Conley index theory).
3. Conditional Energetic Stability: It establishes that rotating circles are conditionally energetically stable. Crucially, it demonstrates that stability holds regardless of the magnitude of the vorticity (Modified Bond Number), provided the perturbations respect the conservation of volume and barycenter constraints. This resolves the apparent instability suggested by the linear analysis of the $(0,0)$ and $(1, \pm 1)$ modes.
4. Generalization of Rayleigh's Work: The results extend Lord Rayleigh's classical analysis of capillary jets (which assumed zero vorticity) to the case of constant non-zero vorticity, confirming that rotational effects do not induce the same instability as translational effects but rather produce oscillatory modes.

4. Main Results

• Theorem (Existence): For specific values of the rotation frequency $\omega$ determined by the dispersion relation, there exist non-trivial rotating wave solutions with $\kappa$-fold symmetry. These solutions bifurcate from the circular state.
  • If the bifurcation occurs from a double eigenvalue, a unique orbit of solutions exists.
  • If it occurs from a quadruple eigenvalue, at least two distinct orbits of solutions exist.
• Theorem (Stability): The rotating circle solution is conditionally energetically stable in the space $H^1 \times H^{1/2}_0$. Specifically, for any $\epsilon > 0$, there exists a $\delta > 0$ such that if the initial data is within $\delta$ of the circle and satisfies the constraints $V = V_{circle}$, $B = 0$, and $P = 0$, the solution remains within $\epsilon$ for all time.
• Linear Stability: The rotating circle is linearly stable (all eigenvalues are purely imaginary).

5. Significance

This work bridges a gap in the mathematical theory of free boundary problems for fluids with vorticity.

• Theoretical Rigor: It moves beyond formal computations (like those of Rayleigh) to provide rigorous existence and stability proofs using modern tools from nonlinear functional analysis and Hamiltonian dynamics.
• Physical Insight: It clarifies the role of vorticity in capillary drops. While vorticity introduces complexity in the Hamiltonian structure and the bifurcation spectrum, it does not necessarily destabilize the circular shape if the physical constraints (volume and center of mass) are maintained.
• Methodological Advancement: The successful application of bifurcation theory to a system with multiplicity 4 eigenvalues and the use of topological methods (Conley blocks) for cases where standard ellipticity fails represent significant technical achievements.
• Relevance: The results are relevant to understanding the stability of liquid drops in microgravity, rotating machinery, and plasma physics where capillary forces and vorticity coexist.

In summary, La Scala provides a complete mathematical framework for 2D capillary drops with constant vorticity, proving that while the system is complex, stable rotating configurations exist and are robust under physically natural constraints.