On the stationary solutions and inviscid limit for the generalized Proudman–Johnson equation with O(1) forcing

Abstract

We consider the stationary problem for the generalized Proudman–Johnson equation ψ ψ x x x − a ψ x ψ x x = ν ψ x x x x + f , where a ∈ R and ν > 0 are parameters and ψ and f satisfy the periodic boundary conditions on [ − π , π ] . We establish the existence of a time-dependent solution and the uniqueness of the stationary solution for some large ν . Then, we investigate the behavior of solution sequences in the limit ν → 0 + , fixing a and f , where bifurcations appear and new solutions emerge. In particular, we analyze if there is convergence towards stationary solutions for the corresponding inviscid equation.