Khakim Egamberganov | Searcharxiv

arXiv2026

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

For $d\geq 4$, we consider incompressible Euler flows in $\mathbb{R}^{d}$ with bi-rotational symmetry and without swirl. Our first result gives the local wellposedness of the Yudovich-type solution. The second result provides global wellposedness up to $d\leq 6$. In particular, it shows that the rate of growth of the vorticity maximum coincides with the rate from axisymmetric flows without swirl, which was obtained in the paper by the second author and Jeong (Arch. Ration. Mech. Anal. 249(3):32, 2025) and Shao--Wei--Zhang (Acta Math. Sin. (Engl. Ser.), 42(3):663-679, 2026).

math.AP↗

arXiv2025

Growth estimates for axisymmetric Euler equations without swirl

We consider the axisymmetric Euler equations in $\mathbb{R}^3$ without swirl, and establish several upper and lower bounds for the growth of solutions. On the one hand, we obtain an upper bound $t^2$ for the radial moment $\int_{\mathbb{R}^3} rω^θdx$, which is the conjectured optimal rate by Childress (Phys. D 237(14-17):1921-1925, 2008). On the other hand, for all initial data satisfying certain symmetry and sign conditions, we prove that the radial moment grows at least like $t/\log t$ as time goes to infinity, and $\|ω(\cdot,t)\|_{L^p(\mathbb{R}^3)}$ exhibits at least $t^{1/4}$ growth in the limsup sense for all $1\leq p\leq \infty$. To the best of our knowledge, this is the first result to establish power-law $L^p$-norm growth for smooth, compactly supported initial vorticity in $\mathbb{R}^3$. For these initial data, we also show that nearly all vorticity must eventually escape to $r\to\infty$ in the time-integral sense.

math.AP↗