On the existence of convex classical solutions to a generalized Prandtl-Batchelor free-boundary problem. II (Q1865585)

For Part I, cf. [the author, ibid. 49, 1-30 (1998; Zbl 0903.35093)]. We give an analytical proof of the existence of convex classical solutions for the (convex) Prandtl-Batchelor free boundary problem in fluid dynamics. In this problem, a convex vortex core of constant vorticity \(\mu>0\) is embedded in a closed irrotational flow inside a closed, convex vessel in \(\mathbb{R}^2\). The unknown boundary of the vortex core is a closed curve \(\Gamma\) along which \((v^+)^2-(v^-)^2=\Lambda\), where \(v^+\) and \(v^-\) denote, respectively, the exterior and interior flow-speeds along \(\Gamma\), and \(\Lambda\) is a given constant. Our existence results all apply to the natural multidimensional mathematical generalization of the above problem. The present existence theorems are the only ones available for the Prandtl-Batchelor problem for \(\Lambda>0\), because (a) the author's prior existence treatment was restricted to the case where \(\Lambda<0\), and because (b) there is no analytical existence theory available for this problem in the nonconvex case, regardless of the sign of \(\Lambda\).