On stationary solutions and inviscid limits for generalized Constantin–Lax–Majda equation with O(1) forcing

Abstract

Abstract The generalized Constantin–Lax–Majda (gCLM) equation was introduced to model the competing effects of advection and vortex stretching in hydrodynamics. Recent investigations revealed possible connections with the two-dimensional turbulence. With this connection in mind, we consider the steady problem for the viscous gCLM equations on T : a v ω x − v x ω = ν Δ ω + f , v = ( − Δ ) − 1 2 ω , where a ∈ R is the parameter measuring the relative strength between advection and stretching, ν > 0 is the viscosity constant, and f is a given O(1)-forcing independent of ν. For some range of parameters, we establish existence and uniqueness of stationary solutions. We then numerically investigate the behaviour of solutions in the vanishing viscosity limit, where bifurcations appear, and new solutions emerge. When the parameter a is away from [−1/2, 1], we verify that there is convergence towards smooth stationary solutions for the corresponding inviscid equation. Moreover, we analyse the inviscid limit in the fractionally dissipative case, as well as the behaviour of singular limiting solutions.