Non-self-similar collapse of three point vortices in a generalized two-dimensional fluid system

Abstract In this study, we investigate the non-self-similar collapse of three vortices in a point vortex model of a generalized two-dimensional (2D) fluid system. In the classical point vortex model, the existence of the self-similar collapse of three vortices has been well known from Gröbli (1877 Vierteljahresschr. Naturforsch. Ges. Zürich 22 129). Moreover, the non-existence of the non-self-similar collapse of three vortices in the classical point vortex model was proved by Hernández–Garduño and Lacomba (2007 J. Math. Fluid Mech. 9 75) and Hiraoka (2008 Nonlinearity 21 361). In this study, we apply a method similar to that of former authors to the point vortex model of the generalized 2D fluid system, which includes a real parameter α , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . We present proofs for the possible existence of the non-self-similar collapse of three vortices for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and for the non-existence of such collapse for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . Furthermore, as an application, we analyze in detail the self-similar and non-self-similar collapses of three point vortices in the symmetric case of the surface quasi-geostrophic system ( α = 1). A close relationship between the self-similar and non-self-similar collapses of the three vortices is identified.