Determine the hysteretic zero-diffusivity limit for non-unique incoming boundary-layer profiles

Determine whether the hysteresis arising from multiple dynamically stable solutions of the nonlinear incoming ODE produces an inviscid run-and-tumble problem whose dynamic boundary condition retains memory.

Background

For nonlinear tumbling laws such as the saturated response r(u) = r_0 + r_1u^2/(1+\zeta u^2), the incoming boundary-layer ODE may have multiple solutions for the same boundary-mass parameter q. The paper reports numerical saddle-node bifurcations and an intermediate regime in which two solutions are stable, creating the possibility of phase transitions and hysteresis in the boundary layer.

The unresolved issue is how this non-uniqueness affects the zero-diffusivity limit. In particular, the authors suggest that the history-dependent selection of stable boundary-layer states could induce memory in the limiting dynamic boundary condition, but they do not establish such a limiting formulation.