Large-data global well-posedness for repulsive singular interactions beyond the threshold

Determine the maximal range of exponents $\alpha\ge3$ in dimensions $d\ge4$ for which the repulsive Vlasov equation with power-law interaction admits global well-posedness for arbitrary large initial data.

Background

The main theorem establishes global well-posedness for both attractive and repulsive interactions only when 0<α<30<\alpha<3. For attractive interactions with α≥3\alpha\ge3, the paper gives a virial obstruction showing finite-time breakdown for negative-energy solutions. That obstruction does not apply to repulsive interactions, for which only small-data global results beyond the threshold are cited. The paper consequently leaves open how far large-data global well-posedness extends in the repulsive case.

References

The second open direction concerns repulsive interactions with $\alpha\ge3, \qquad d\ge4$. The virial obstruction in Theorem~\ref{thm:virial-obstruction} is specific to the attractive sign and gives no corresponding mechanism for breakdown in the repulsive problem. There is already small--data global theory beyond the threshold $\alpha=3$. For the repulsive Vlasov--Poisson equation in dimensions d\ge4, corresponding in our notation to the Coulomb exponent \alpha=d-1\ge3, Pankavich constructed global solutions under a smallness condition on suitable integrated moments. He also established scattering and precise asymptotic behavior under additional decay assumptions. This suggests that large-data global well-posedness may persist beyond the range proved here. Determining the maximal range of singular repulsive potentials for which large-data global well-posedness holds appears to be a natural next step.

— Global well-posedness for Vlasov equations with power-law interactions in $d\ge4$  (2609.09512 - Katriadakis, 8 Sep 2026) in Section 1, subsection “Open problems”