$L^p$ Stability of Vortex Patches in Two Dimensional Domains

Technical Summary: $L^p$ Stability of Vortex Patches in Two-Dimensional Domains

Problem Statement
This paper investigates the orbital stability of vortex patches governed by the two-dimensional incompressible Euler equations on fixed domains $\Omega \subseteq \mathbb{R}^2_+$. The vorticity $\omega$ evolves according to $\partial_t \omega + u \cdot \nabla \omega = 0$ with slip boundary conditions. While general solutions to the 2D Euler equations can exhibit instability (e.g., double-exponential growth of vorticity gradients), stability can be established for specific classes of initial data using variational methods.

The specific challenge addressed is the lack of a unified $L^p$ stability theory for vortex patches on general two-dimensional domains that operates without a priori $L^1$ or $L^p$ bounds on the vorticity. Previous works have been limited by specific constraints:

• Abe and Choi [1] established stability for Lamb dipoles in the half-plane but required an $L^1$ bound.
• Abe, Choi, and Jeong [2] removed the $L^1$ constraint but were restricted to the $L^2$ setting.
• Dong and Luo [9] extended results to strips and weak finite volume domains but retained the $L^1$ constraint and were limited to $L^2$ stability.

Methodology
The author employs a penalized energy variational framework. They define the penalized energy functional:
$$E_{p,\lambda}[\omega] = E[\omega] - \frac{1}{p}\lambda \|\omega\|_p^p$$
where $E[\omega]$ is the kinetic energy and $\lambda > 0$ is a parameter. The study focuses on the minimization problem:
$$I_{\mu,\lambda}(p) = \inf_{\omega \in K_\mu} \{-E_{p,\lambda}[\omega]\}$$
where the admissible space $K_\mu$ consists of non-negative vorticities with a fixed first moment (impulse) $\int_\Omega x_2 \omega \, dx = \mu$, but crucially, without an imposed $L^1$ or $L^p$ upper bound.

The core methodology involves three main technical components:

1. Energy Estimates and Parameter Ranges: Establishing uniform $L^p$ bounds for minimizing sequences using Hölder's inequality and Young's inequality, identifying critical thresholds for $p$ (specifically $p > 4/3$) and conditions on $\lambda$ to ensure the infimum is negative.
2. Existence of Minimizers: Proving the existence of minimizers and their compactness. This requires overcoming the loss of spatial scaling and translation invariance in general domains.
   • For the Half-Plane ($\mathbb{R}^2_+$): Scaling arguments are used to reduce the problem to a standard form.
   • For Strips ($S$): A refined concentration-compactness argument is developed. They utilize Steiner symmetrization to prove strict subadditivity of the energy functional, which excludes the "dichotomy" case in Lions' compactness principle.
   • For Weak Finite Volume Domains ($D$): They bypass the need for strict subadditivity by exploiting the inherent decay rate $q$ of the domain's geometry (specifically the volume condition $\int_{D_l} x_2^q dx < \infty$) to enforce compactness directly.
3. Orbital Stability: Demonstrating that the set of minimizers is orbitally stable under Eulerian dynamics. This relies on the conservation of $L^p$ norms and the first moment along particle trajectories, combined with the general convergence theorem for minimizing sequences.

Key Contributions and Results
The paper establishes a unified $L^p$ stability theory for three specific types of domains: the half-plane, strips of any width, and domains satisfying a weak finite volume condition.

• Existence and Compactness (Theorem 1.4): The author proves that for suitable parameters $p, \mu, \lambda$, the penalized energy functional admits a minimizer. Every minimizing sequence possesses a subsequence that converges strongly in $L^p$ and $L^1$ (for the weighted moment) to a minimizer $\omega$.
  • For $\mathbb{R}^2_+$: Valid for $p > 4/3$.
  • For Strips: Valid for $p > 2$ with any $\lambda > 0$, or for $p = 2$ provided $\lambda > 2\pi^4/L^2$.
  • For Weak Finite Volume Domains: Valid for $p > 4/3$ if the decay rate $q \in [0, 2]$, and for $4/3 < p \le q/(q-1)$ if $q \in (2, 4)$.
• Structural Properties: Minimizers satisfy the elliptic equation $\omega^{p-1} = \lambda(\psi - Wx_2)_+$, where $\psi$ is the stream function and $W$ is a strictly positive constant. Minimizers are shown to have compact support and possess specific regularity ($C^{m,\alpha}$).
• Orbital Stability (Theorem 1.5): The set of minimizers $S_{\mu,\lambda}$ is orbitally stable. For any $\epsilon > 0$, there exists a $\delta > 0$ such that if the initial data is within $\delta$ of the set of minimizers (in terms of $L^p$ distance and weighted $L^1$ distance), the solution remains within $\epsilon$ of the set for all time.
• Extension with $L^1$ Constraint (Theorem 1.7): As a supplement, the paper provides stability results for the weak finite volume domain $D$ for all $q \ge 0$ and $p > 1$, provided an $L^1$ upper bound is imposed on the admissible space.

Significance
The paper claims to synthesize and extend the approaches of Abe and Choi [1], Abe et al. [2], and Dong and Luo [9] into a comprehensive framework. Its primary significance lies in:

1. Removing Constraints: It successfully removes the a priori $L^1$ bound requirement for stability in the half-plane and strip cases, and extends $L^2$ results to general $L^p$ ($p > 4/3$) without such bounds.
2. Unified Framework: It provides a unified theory covering three distinct geometric settings (half-plane, strip, and general weak finite volume domains) that were previously treated separately or with limitations.
3. Novel Strategies: The work introduces novel strategies to handle the loss of scaling and translation invariance. Specifically, the use of domain decay rates to enforce compactness in weak finite volume domains and the refined Steiner symmetrization arguments for strips represent significant methodological advancements over previous concentration-compactness applications.

The author notes that while the results significantly expand the scope of provably stable vortex structures, limitations remain, particularly regarding the critical threshold $p=4/3$ and the necessity of $L^1$ bounds for certain parameter ranges in general domains. The paper concludes by identifying open questions regarding optimal geometric conditions, free-boundary problems, and the uniqueness and finer structure of minimizers.