Post-catastrophe singular limit and uniqueness for the non-conservative non-local Burgers approximation

Determine whether, for the non-conservative, non-local Burgers regularization ∂_t u^ε + (η_ε ∗ u^ε) ∂_x u^ε = 0 with piecewise Lipschitz non-decreasing initial data featuring entropic jump discontinuities at finitely many points, the limit as ε → 0 after the time when the discontinuity curves interact (the secondary catastrophe time) coincides with the entropy solution of the local Burgers equation ∂_t u + ∂_x(u^2/2) = 0, and ascertain whether this ε → 0 limit is unique.

Background

The paper introduces a non-conservative, non-local approximation to the Burgers equation and proves convergence to the entropy solution up to the time when characteristics remain non-intersecting, including for certain piecewise Lipschitz increasing data with entropic (downward) jump discontinuities. The authors define a finite 'secondary catastrophe' time beyond which discontinuity curves (shocks) may interact, at which point their convergence proof no longer applies.

Beyond this interaction time, prior results (e.g., Coron et al.) ensure that every converging subsequence of non-local solutions yields a weak solution to the local conservation law, but the identification of this limit with the entropy solution and the uniqueness of the limit remain unresolved in the present framework. Establishing these properties would complete the justification of the singular limit for broader classes of discontinuous initial data.