On the swirling flow between rotating coaxial disks: Existence and nonuniqueness

Consider solutions G(x,epsilon), H(x,epsilon)> of the von Karman equations for the swirling flow between two rotating coaxial disks (1.1) epsilonH superscript iv + HH''' + GG' equal 0 and (1.2) epsilonG' + HG' H'G equal 0 with boundary conditions (1.3 H(0,epsilon) equal H' (0,epsilon) equal H(1, epsilon) equal H'(1, epsilon) equal 0 (1.4) G(0, epsilon) equal s, G(1, epsilon) equal 1, s < 1. In this work we establish the existence of solutions for epsilon small enough. In fact, if n is a given positive integer with sign s equal (-1 to the n power) then there is - for epsilon greater than 0 sufficiently small - a solution with the additional property: G(x, epsilon) has n interior zeros. If n 1 there are at least two such solutions. If s equal 0 there is at least one such solution for every positive integer n. The asymptotic shape of these solutions is described.