2024/09/29 by Kyudong Choi, In-Jee Jeong, Choi, Kyudong +3 · 5 citations
Engineering · Physics and Astronomy · #Superconducting Materials and Applications #Physics of Superconductivity and Magnetism #Magnetic confinement fusion research
paper · pdf · doi:10.48550/arxiv.2409.19822
For the 2D incompressible Euler equations, we establish global-in-time (t ∈ ℝ) stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is signed, sufficiently concentrated and close to its radial rearrangement up to a translation in L1, we prove that it remains so for all times. The main difficulty is that the kinetic energy maximization problem in a quadrant -- the typical approach for establishing vortex stability -- lacks a solution, as the kinetic energy continues to increase when the vorticity escapes to infinity. We overcome this by taking dynamical information into account: finite-time desingularization result is combined with monotonicity of the first moment and a careful analysis of the interaction energies between vortices. The latter is achieved by new pointwise estimates on the Biot--Savart kernel and quantitative stability results for general interaction kernels. Moreover, with a similar strategy we obtain stability of a pair of opposite-signed Lamb dipoles moving away from each other.