Existence of classical critical points for the two-point action

Determine whether a classical critical point of the $L^2_tL^2_x$ action functional exists for every pair of smooth endpoint configurations, even when the initial configuration is the identity and the target configuration is a smooth volume-preserving diffeomorphism.

Background

The paper formulates the Lagrangian two-point boundary-value problem for incompressible fluids as the search for an action-minimizing path of volume-preserving diffeomorphisms between prescribed endpoint configurations. Although generalized incompressible flows provide minimizers in a broader nonsmooth setting, the existence of classical critical points remains unresolved even for smooth endpoints. The authors distinguish this issue from Shnirelman's nonexistence result for minimizers in dimensions three and higher, which does not itself rule out classical critical points.

References

Even if the initial state $\varphi_0 = id$ and the target configuration $\varphi_T=f$ are both smooth, it is unknown whether a critical point to intro:energy exists in a classical sense.

— 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

Let $f \in \text{SDiff}([0,1]2)$ and assume there exists a path ${\varphi_t} \subset \text{SDiff}([0,1]2)$ connecting $id$ to $f$ such that $\mathcal{A}{\varphi_t} < +\infty$. Then, there exists a minimizer ${\bar\varphi_t}$ connecting $id$ to $f$ such that

— 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 “Shnirelman’s two-point conjecture,” Conjecture 1

Let $f \in \text{SDiff}([0,1]2)$ and assume there exists ${\varphi_t} \subset \text{SDiff}([0,1]2)$ connecting $id$ to $f$ with $\mathcal{A}{\varphi_t} < +\infty$. Then, there exists a smooth solution $u: [0,1] \times [0,1]2 \to \mathbb{R}2$ of the Euler equations such that $\varphiu_{t=1} = f$, where $\varphiu$ denotes the flow of the vector field $u$.

— 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 “Shnirelman’s two-point conjecture,” Conjecture 2