On the upper bound for the vorticity growth of bi-rotational Euler flows without swirl

Technical Summary: Upper Bound for Vorticity Growth of Bi-Rotational Euler Flows Without Swirl

Problem Statement
This paper investigates the incompressible Euler equations in $\mathbb{R}^d$ ($d \ge 4$) under the assumption of bi-rotational symmetry without swirl. The velocity field $v$ is restricted to the form $v = u_r(r,s)e_r + u_s(r,s)e_s$ in bi-polar coordinates $(r, s)$, where $r^2 = \sum_{i=1}^{n+1} x_i^2$ and $s^2 = \sum_{i=n+2}^{n+m+2} x_i^2$ (with $d = n+m+2$). The primary objective is to establish the well-posedness of solutions and to derive precise upper bounds on the growth rate of the vorticity maximum $\|\omega\|_{L^\infty}$ over time. Specifically, the authors aim to determine if the vorticity growth in these high-dimensional bi-rotational flows mimics the behavior observed in axisymmetric flows without swirl in $\mathbb{R}^3$.

Methodology
The analysis relies on a reduction of the Euler equations to a scalar transport equation for the scalar vorticity $w$. The key methodological components include:

1. Scalar Formulation: The vorticity tensor is reduced to a scalar quantity $w = \partial_r u_s - \partial_s u_r$. The evolution of $w$ is governed by a transport equation with a source term representing vortex stretching:
   $$\partial_t w + (u_r \partial_r + u_s \partial_s)w = \left(\frac{n u_r}{r} + \frac{m u_s}{s}\right)w.$$
   Crucially, the quantity $w/(r^n s^m)$ is conserved along particle trajectories.

2. Local Well-posedness (Yudovich-type): The authors establish local well-posedness for initial data $w_0$ in the intersection of Lorentz space $L^{d,1}$ and $L^\infty$, with specific weighted decay conditions. This is achieved by proving that the Biot–Savart kernel for bi-rotational flows maps the dual space of $L^{d,1}$ to $L^\infty$, allowing for standard energy estimates.

3. Global Well-posedness and Growth Bounds ($d \le 6$): For dimensions $d \le 6$, the authors prove global existence and derive growth rates under additional assumptions on the initial data (finite kinetic energy and specific decay near the axes). The proof strategy involves:

   • Conservation Laws: Utilizing the conservation of $w/(r^n s^m)$ to bound the vorticity maximum by the time integral of the velocity maximum.
   • Feng–Šverák-type Estimates: Deriving a new estimate for the velocity maximum $\|u\|_{L^\infty}$ in terms of the $L^\infty$ norm of the relative vorticity $w/(r^n s^m)$ and two radial moments: $\|r^m w/s^n\|_{L^1}$ and $\|s^n w/r^m\|_{L^1}$. This is achieved by analyzing the decay properties of the bi-rotational Biot–Savart kernel, expressing it via single-variable functions to obtain appropriate decay rates.
   • Radial Moment Evolution: Establishing differential inequalities for the time evolution of the radial moments. By combining these with the velocity estimates, the authors derive a closed differential inequality for the sum of the moments.
   • ODE Analysis: Solving the resulting differential inequality to determine the time dependence of the velocity and subsequently the vorticity.

Key Results

• Theorem 1.1 (Local Well-posedness): For any $d \ge 4$, there exists a unique local-in-time solution for initial data $w_0 \in (L^{d,1} \cap L^\infty)(\mathbb{R}^d)$ satisfying specific weighted integrability conditions. The solution remains in the same regularity class for a short time interval.

• Theorem 1.2 (Global Well-posedness and Growth Rates): For dimensions $d \le 6$, assuming the initial data satisfies finite kinetic energy and specific decay conditions (including $w_0/(r^n s^m) \in L^1 \cap L^\infty$), the solution exists globally in time. The paper establishes the following upper bounds for the vorticity maximum $\|w(t)\|_{L^\infty}$:

  • $d=4, 5$: Polynomial growth of order $(1+t)^{4(d-2)/(6-d)}$. Specifically, $O((1+t)^4)$ for $d=4$ and $O((1+t)^{12})$ for $d=5$.
  • $d=6$: Exponential growth $O(e^{Ct})$.

Significance and Claims
The paper claims that the derived growth rates for bi-rotational flows in dimensions $d=4, 5, 6$ coincide exactly with the upper bounds previously established for axisymmetric flows without swirl in $\mathbb{R}^d$ (referencing Jeong–Lim and Shao–Wei–Zhang).

The authors highlight that their work extends the understanding of vortex stretching mechanisms to higher dimensions with bi-rotational symmetry. A central technical contribution is the successful simplification of the bi-rotational Biot–Savart kernel, allowing for the derivation of the necessary decay estimates that were previously difficult to obtain. This enables the extension of the "Childress dipole model" intuition to higher dimensions, demonstrating that the mechanism driving vorticity growth (the expansion of fluid particles away from the rotation axes) yields the same scaling laws in the bi-rotational setting as in the axisymmetric case. The results suggest that for $d \le 6$, the vortex stretching is not strong enough to cause finite-time blow-up, although the growth can be rapid (polynomial or exponential).