Mixed Uniqueness and Non-Uniqueness at Common Positive-Time Regularity

Determine whether there exists a one-dimensional hyperbolic system of conservation laws satisfying natural assumptions for which some initial data produce unique solutions and other initial data produce non-unique solutions, even though all solutions under consideration have the same degree of regularity at positive times.

Background

The main theorem establishes uniqueness under the Strong Trace Property and non-uniqueness for solutions that may lack it, using simple Riemann initial data. The authors then ask whether a comparable distinction can occur for lower-regularity initial data while all competing solutions share the same positive-time regularity.

This problem is linked in the paper to Bressan’s Open Problem #6 and to efforts to construct globally continuous convex-integration solutions for one-dimensional systems.

References

Does there exist a hyperbolic system of conservation laws in one spatial dimension for which under natural assumptions, some initial data yield unique solutions and some initial data yield non-unique solutions -- even when all solutions under consideration have the same degree of regularity at positive times?

— The computational ansatz for convex integration of hyperbolic systems and a resolution of the Strong Trace Conjecture  (2609.09353 - Krupa, 8 Sep 2026) in Section 1, subsection “Further questions on rank-one geometry,” subsubsection “Open questions”