Search for: All records

Abstract The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation $-{\mathrm{\Lambda }}^{\sigma }$, where $\widehat{{\mathrm{\Lambda }}^{\sigma }}=|k{|}^{\sigma }$, both for the problem on the circle $x\in [-\pi ,\pi ]$and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\sigma \text{⩾}1$and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when $a=1/2$, for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory.