2022/07/13 by T. Cieślak, Cieślak, T., P. Kokocki +3 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2207.06056
openalex publication_date 2022/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider solutions of the 2D incompressible Euler equation in the form of M≥ 1 cocentric logarithmic spirals. We prove the existence of a generic family of spirals that are nonsymmetric in the sense that the angles of the individual spirals are not uniformly distributed over the unit circle. Namely, we show that if M=2 or M≥ 3 is an odd integer such that certain non-degeneracy conditions hold, then, for each n ∈ \ 1,2 \, there exists a logarithmic spiral with M branches of relative angles arbitrarily close to θk = knπ/M for k=0,1,… , M-1, which include halves of the angles of the Alexander spirals. We show that the non-degeneracy conditions are satisfied if M∈ \ 2, 3,5,7,9 \, and that the conditions hold for all odd M>9 given a certain gradient matrix is invertible, which appears to be true by numerical computations.