The computational ansatz for convex integration of hyperbolic systems and a resolution of the Strong Trace Conjecture

RESUMO

In this paper, we consider $2\times2$ hyperbolic systems of conservation laws in one spatial dimension. We use the computational ansatz introduced in the hyperbolic theory by the author and Székelyhidi to study the constitutive set corresponding to the PDE. The rank-one convex geometry of this set relates to non-uniqueness and the existence of low-regularity solutions. Through a computer-assisted search, we find a pressure law $p$ such that the $p$-system with this pressure law, and its natural strictly convex entropy, verifies all of the conditions necessary for the large data $L^2$ stability and the technique of ``$a$-contraction with shifts'' and thus we have uniqueness of certain Riemann solutions in the class of solutions verifying the Strong Trace Property. At the same time, the constitutive set contains a $T_6$ configuration and we use it to construct non-unique solutions without the Strong Trace Property. This resolves the question of sharpness of strong traces. We also present proofs which show nonexistence of $T_\infty$ structures for all genuinely nonlinear systems and nonexistence of $T_N$ structures for all $N$ for the $p$-system with $p''>0$, thus blocking these routes towards convex integration.