Origin and generality of irreversible mixing behavior

Determine whether the irreversible behavior produced by mixing in the two-point boundary-value formulation of two-dimensional incompressible flow also occurs in the Cauchy problem formulation of the Euler equations, rather than being specific to the two-point formulation.

Background

The paper constructs relaxed two-dimensional flows in which trajectories entering an intersection region lose information about their initial configuration, producing an apparently irreversible mixing behavior. The authors relate this phenomenon to generalized incompressible flows and explicitly identify its dependence on the formulation of the problem as unresolved. The question is whether the same behavior should be expected for Euler evolution from prescribed initial data.

References

Secondly, it remains an open question to understand whether this behavior originates from the two-point boundary value formulation or if it is also expected in the Cauchy problem formulation of Euler.

— On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation  (2609.31294 - Schiffer et al., 25 Sep 2026) in Section 2, Subsection “Connections to Generalized Flows”

We point out that this behavior has been observed numerically in and we do not know if it is merely related to the two-point boundary value formulation of Euler Equations, or it can be used to partially explain the dynamics of Euler Equations in $2d$.

— On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation  (2609.31294 - Schiffer et al., 25 Sep 2026) in Section 1, Introduction