Self-Similar Solutions of the Two-Dimensional Incompressible Euler Equation from Large Initial Data

Technical Summary: Self-Similar Solutions of the Two-Dimensional Incompressible Euler Equation from Large Initial Data

Problem Statement
The paper addresses the construction of forward-in-time self-similar solutions for the two-dimensional incompressible Euler equations:
$$\partial_t u + u \cdot \nabla u + \nabla p = 0, \quad \text{div } u = 0$$
in $\mathbb{R}^2 \times (0, \infty)$. The study focuses on initial data $u_0$ that is $C^1$, divergence-free, and $(-a)$-homogeneous, where the scaling parameter satisfies $1/3 < a < 1$. Specifically, the initial velocity takes the form $u_0(x) = |x|^{-a}u_0(x/|x|)$.

The central challenge is to establish the existence of a global weak solution $u(t,x)$ that preserves this self-similar structure:
$$u(t, x) = t^{-\frac{a}{1+a}} U\left( \frac{x}{t^{\frac{1}{1+a}}} \right)$$
without imposing smallness conditions on the initial data or sign restrictions on the vorticity. This problem sits at the intersection of existence theory, singularity formation, and the potential non-uniqueness of weak solutions. While self-similar solutions for the Navier-Stokes equations (both 2D and 3D) have been constructed for large data, the Euler case (zero viscosity) presents distinct difficulties, particularly regarding the lack of dissipation to control the vorticity and the critical nature of the scaling.

Methodology
The construction proceeds via a vanishing-dissipation limit of hypodissipative profiles. The authors introduce a fractional dissipation term $\epsilon \Lambda^{1+a}$ (where $\Lambda = (-\Delta)^{1/2}$) and consider the regularized equation:
$$\epsilon \Lambda^{1+a} U_\epsilon - \frac{1}{1+a}(a U_\epsilon + y \cdot \nabla U_\epsilon) + U_\epsilon \cdot \nabla U_\epsilon + \nabla P_\epsilon = 0$$
The strategy involves three main steps:

1. Existence of Hypodissipative Profiles: For any fixed $\epsilon > 0$, the existence of a smooth profile $U_\epsilon$ is established. The authors provide an independent proof of existence (building on work by Hou and Song) using a divergence-free far-field cutoff and a Galerkin scheme. Crucially, they derive pointwise decay estimates for the profile and its derivatives directly from Oseen-kernel estimates, avoiding weighted energy spaces.
2. Uniform Critical Vorticity Estimate: The core of the paper is establishing an estimate for the vorticity $\Omega_\epsilon = \text{curl } U_\epsilon$ that is uniform as $\epsilon \to 0$. The authors define the critical exponent $p_c = 2/(1+a)$. By testing the vorticity equation against the derivative of the truncation function $(|\Omega_\epsilon| - k)_+$, they derive a differential inequality for the distribution function of the vorticity. This yields a uniform bound in the critical Lorentz space $L^{p_c, \infty}(\mathbb{R}^2)$:
   $$\|\Omega_\epsilon\|_{L^{p_c, \infty}} \leq C(a) \|\omega_0\|_{L^{p_c}(S^1)}$$
   This estimate relies on the specific divergence of the "pseudo-velocity" field $\tilde{U}_\epsilon = U_\epsilon - \frac{y}{1+a}$, which equals $-p_c$, allowing for a scale-invariant control of the vorticity truncation.
3. Compactness and Limit Passage: Using the uniform Lorentz bounds on $U_\epsilon$ and $\nabla U_\epsilon$, the authors extract a subsequence converging to a limit profile $(U, P)$. The strong local convergence of the velocity and weak-* convergence of the vorticity allow passing to the limit in the nonlinear terms. For the range $1/3 < a < 1/2$, the product $U \Omega$ is shown to be locally integrable, ensuring the vorticity equation holds in the distributional sense.

Key Results
The main theorem (Theorem 1.2) establishes that for any $C^1$ divergence-free $(-a)$-homogeneous initial datum with $1/3 < a < 1$, there exists a self-similar weak solution $u(t,x)$ with the following properties:

• Regularity: The velocity $u$ is continuous in time with values in $L^2_{\text{loc}}(\mathbb{R}^2)$, and the gradient $\nabla u$ belongs to $L^\infty_t L^{p_c, \infty}_x$.
• Initial Trace: The solution converges to the initial datum $u_0$ strongly in $L^2_{\text{loc}}$ as $t \to 0$. The vorticity $\omega(t)$ converges weak-* to $\omega_0$ in $L^{p_c, \infty}$.
• Generality: The construction requires no smallness assumption on the initial data and no sign assumption on the vorticity. The angular profile $u_0$ need not be close to a radial vortex or satisfy a radial dominance condition, distinguishing this result from previous spiral constructions.
• Endpoint Behavior: The result is subcritical with respect to the endpoint $a=1$. At $a=1$, the favorable factor in the vorticity truncation identity vanishes, and the argument does not exclude concentration measures.

Significance and Claims
The paper positions its contribution within the broader context of fluid mechanics, specifically regarding the non-uniqueness of weak solutions and the formation of singularities.

• Support for Non-Uniqueness: The authors state that the existence of arbitrarily large self-similar profiles provides a necessary "base state" for future investigations into instability and bifurcation scenarios, which are central to the non-uniqueness program (referencing works by Jia, Šverák, and others). While this paper does not prove non-uniqueness itself, it furnishes the class of large solutions required for such an analysis.
• Extension of Spiral Solutions: Unlike previous constructions of algebraic spiral solutions which relied on high-frequency symmetry or perturbations of radial vortices, this work handles general $C^1$ angular data without symmetry assumptions.
• Critical Control: The result highlights that global weak existence can be achieved with control in the critical Lorentz space $L^{p_c, \infty}$ rather than requiring bounded vorticity ($L^\infty$) or finite total vorticity. This extends the known existence theory for rough data classes in two-dimensional ideal flow.

The paper concludes by noting that for $a \geq 1/2$, the vorticity equation holds in the sense of distributions only if $U\Omega \in L^1_{\text{loc}}$, which is not guaranteed by the estimates in that range, though the velocity formulation remains valid. The construction is explicitly described as a "vanishing-dissipation limit," distinguishing it from direct construction methods.