arXiv subjects

Averkios Averkiou

Publications and source records attributed to Averkios Averkiou.

3 recordsLinked to original sources

arXiv2026

An exact Delaunay flux identity for compact equilibria in the liquid drop model

We revisit the construction of compact liquid drop equilibria from Delaunay unduloids in [Arch. Ration. Mech. Anal. 249 (2025), Paper No. 74]. We evaluate exactly the coefficient arising in the final solvability condition of the Lyapunov-Schmidt reduction, identifying it as the product of the axial period and the normalised Delaunay flux. Using this identity and the monotonicity of the period, we establish the conjectured positivity of the coefficient and prove that its product with the period strictly increases with the neck radius. We also derive an exact relation between the volume and the surface area of one period and prove that both strictly decrease with the neck radius. These results extend the construction throughout the embedded unduloid family and yield compact equilibria of every sufficiently large mass, with reference neck radii converging to any prescribed interior value as the mass tends to infinity.

math.AP↗

arXiv2026

Clustered vortex helices with compactly supported cross-sectional vorticity in the 3D Euler equations

We consider the three-dimensional incompressible Euler equations for helical flows without swirl. By adapting gluing techniques, we construct the first smooth multi-vortex solution in the whole space $\mathbb{R}^3$ exhibiting a cluster of collapsing helical filaments, with the associated cross-sectional vorticity remaining compactly supported in $\mathbb{R}^2$ for all times. Our result generalises previous collapsing configurations in $\mathbb{R}^3$ with rapidly decaying vorticity cores, and extends related variational solutions obtained in infinite cylindrical domains.

math.AP↗

arXiv2025

Helical vortex filaments with compactly supported cross-sectional vorticity for the incompressible Euler equations in $\mathbb{R}^3$

We revisit the vortex filament conjecture for three-dimensional inviscid and incompressible Euler flows with helical symmetry and no swirl. Using gluing arguments, we provide the first construction of a smooth helical vortex filament in the whole space $\mathbb{R}^3$ whose cross-sectional vorticity is compactly supported in $\mathbb{R}^2$ for all times. The construction extends to a multi-vortex solution comprising several helical filaments arranged along a regular polygon. Our approach yields fine asymptotics for the vorticity cores, thus improving related variational results for smooth solutions in bounded helical domains and infinite pipes, as well as non-smooth vortex patches in the whole space.

math.AP↗