Strong Trace Property for General Conservation-Law Solutions

Determine whether every bounded entropy solution of a one-dimensional hyperbolic system of conservation laws satisfies the Strong Trace Property, including solutions constructed by the compensated compactness method.

Background

The paper’s L2L^2 stability and uniqueness theory relies on the Strong Trace Property, which controls the behavior of solutions near arbitrary Lipschitz curves and excludes extreme oscillations. Although finite-total-variation and Glimm–Lax solutions are known to possess this property, the authors state that its validity for general solutions remains unresolved.

The question is important because the Strong Trace Property is the regularity threshold separating the uniqueness result proved in the paper from the non-uniqueness constructions obtained without that property.

References

However, in general it is not known if solutions to conservation laws must satisfy the Strong Trace Property. In particular, it is currently unknown if solutions originating from the compensated compactness method verify the Strong Trace Property (however, see some related works on this question ).

— 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 “Strong Trace Property”

Is the Strong Trace Property truly the precise boundary between uniqueness and non-uniqueness? Or are there weaker assumptions that can also ensure uniqueness?

— 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”

However, whether general bounded entropy solutions satisfy the Strong Trace Property used here remains open.

— Weak-BV stability of inflow and outflow problems for the isentropic Euler system  (2610.03236 - Kang et al., 2 Oct 2026) in Section 1, subsection “Literature review”