2009/03/18 by W. Pauls, Walter Pauls · 5 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Boundary (topology) #Domain (mathematical analysis) #Euler's formula #Exponent #Fluid Dynamics and Turbulent Flows #Fourier transform #Gravitational singularity #Mathematical analysis #Mathematical physics #Mathematics #Navier-Stokes equation solutions #Physics #Type (biology) #Zero (linguistics) #nlin.CD
paper · pdf · doi:10.1016/j.physd.2010.03.004
published in Physica D Nonlinear Phenomena 239(13), 1159-1169 (Elsevier BV) · 12 pages, 7 figures
arxiv created 2009/03/18 · openalex publication_date 2010/03/22 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We present a study of complex singularities of a two-parameter family of solutions for the two-dimensional Euler equation with periodic boundary conditions and initial conditions F(p) cos p z + F(q) cos q z in the short-time asymptotic regime. As has been shown numerically in W. Pauls et al., Physica D 219, 40-59 (2006), the type of the singularities depends on the angle between the modes p and q. Here we show for the two particular cases of the angle going to zero and to pi that the type of the singularities can be determined very accurately, being characterised by the values 5/2 and 3 respectively. In these two cases we are also able to determine the subdominant corrections. Furthermore, we find that the geometry of the singularities in these two cases is completely different, the singular manifold being located "over" different points in the real domain.