Self-consistent Euler energy escape and kinetic realization

Construct conservative smooth or vanishing-forcing axisymmetric Euler approximations whose kinetic energy escapes to spatial infinity in the weak limit, and determine whether such an approximation family can be lifted to collisional dynamics to produce a renormalized Boltzmann solution with a strict global energy inequality.

Background

The paper constructs a passive-transport obstruction below the moment-free threshold p=4/3: transported axisymmetric scalar data can have vanishing L1\cap Lp norms and vanishing initial Biot–Savart energy while producing an order-one terminal energy packet at large cylindrical radius. However, the prescribed transport field is not generally the velocity induced by the transported scalar, so the construction is not a self-consistent Euler example.

The authors explicitly leave unresolved whether the same large-radius energy escape can occur for conservative smooth or vanishing-forcing Euler approximations. They further identify a related kinetic question: whether an Euler family exhibiting such escape could be lifted to collisional dynamics yielding a renormalized Boltzmann solution that satisfies a strict global energy inequality.

References

Proposition \ref{prop:passive-shell-obstruction} identifies a passive-transport obstruction below the moment-free $4/3$ threshold, but leaves open whether the same large-radius energy escape can occur self-consistently under the Euler dynamics. More precisely, can one construct conservative smooth or vanishing-forcing Euler approximations whose energy escapes to spatial infinity in the weak limit? A positive answer would also raise the kinetic question of whether such a family can be lifted to collisional dynamics so as to produce a renormalized Boltzmann solution with a strict global energy inequality.

— Rough Axisymmetric Euler Dynamics: Global Energy Compactness and Kinetic Passage Across Regularity Breakdown  (2610.06385 - Bae et al., 5 Oct 2026) in Remark following Proposition 4.?, Section 2 (the remark immediately following Proposition \ref{prop:passive-shell-obstruction})