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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 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.
Caflisch et al. (Wed,) studied this question.
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