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Propagated regularity in dimensions d ≥ 2

Ascertain the regularity that solutions to the nonlocal conservation law ∂_t u − div(u^m ∇g ∗ u) = 0 can propagate in dimensions d ≥ 2, identifying precise function spaces (e.g., continuity, BV, Sobolev) and assumptions on m and initial data under which such regularity persists despite potential shock formation.

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Background

The model can generate shocks when m ≠ 1, and no general comparison principle is available. In one dimension and for special radial cases, regularity mechanisms are better understood via the associated Hamilton–Jacobi formulation. In higher dimensions, however, it is unclear what regularity is maintained over time.

Clarifying regularity propagation is important for stability, numerical analysis, and the validity of weak–strong uniqueness arguments beyond short times.

References

For general dimensions $d\ge 2$, it is not clear which kind of regularity could be propagated by the equation, although some specific solutions give a hint (see the Appendix).

— On a repulsion model with Coulomb interaction and nonlinear mobility (2510.16894 - Courcel et al., 19 Oct 2025) in Subsection “Related works” (Introduction)