Singular Solutions and Ill-Posedness for the Evolution of Vortex Sheets

Abstract

The evolution of a planar vortex sheet is described by the Birkhoff–Rott equation. Duchon and Robert [C.R. Acad. Sci. Paris, 302 (1986), pp. 183–186], [Comm. Partial Differential Equations, 13 (1988), pp. 1265–1295] have constructed exact solutions of this equation that are analytic for all $t < 0$ but have a possible singularity in the curvature of the sheet at $t = 0$. This shows that smooth initial data for a vortex sheet can lead to singularity formation at a finite time, in agreement with the results of numerical computation [J. Fluid Mech., 167 (1986), pp. 65–93], [J. Fluid Mech., 114 (1982), pp. 283–298] and of asymptotic expansion [Proc. Roy. Soc. London Ser A, 365 (1979), pp. 105–119], [Theoretical and Applied Mechanics, in Proc. XVI Internat. Congr. Theoret. Appl. Mech., F. I. Niordson and N. Olhoff, eds., North-Holland, Amsterdam, 1984, pp. 629–633]. We present an independent construction of these solutions and use these results to infer that the vortex sheet problem is ill-posed in Sobolev class $H_n $ with $n > {3 / 2}$. Earlier results show well-posedness in an analytic function class [Comm. Pure Appl. Math., 39 (1986), pp. 807–838], [Comm. Math. Phys., 80 (1981), pp. 485–516]. Our method is to construct an explicit singular function that is a solution of the linearized equation, with a correction term added on to make the sum an exact solution of the nonlinear equation. The correction term is analyzed using the Cauchy–Kowalewski method.