The Riemann problem is the simplest test case in gas dynamics: two uniform states separated by a single discontinuity, and you ask how the system evolves. In one dimension, the answer is classical — a unique solution built from shocks, rarefactions, and contact discontinuities. Every introductory course in computational fluid dynamics starts here because the solution is known, exact, and unique.
In two dimensions, the uniqueness breaks. The paper (arXiv:2603.23921, March 2026) demonstrates that for the two-dimensional barotropic compressible Euler equations, specific Riemann initial data — conditions whose one-dimensional self-similar solution contains only a contact discontinuity — admit infinitely many admissible weak solutions. Not just two or three alternative interpretations. Infinitely many.
The technique is convex integration — a method from the theory of differential inclusions that constructs solutions by iteratively adding fine-scale oscillations. Each oscillation satisfies the equations and the admissibility conditions (entropy inequality), and different sequences of oscillations converge to different solutions. The method exploits the fact that the equations are underdetermined in two dimensions: the Rankine-Hugoniot conditions that pin down one-dimensional solutions leave enough freedom in two dimensions for an infinite family.
The structural lesson is about the relationship between physics and mathematics. The equations are the same. The initial conditions are the same. The admissibility criteria are the same. But the dimensionality of the problem changes the answer from unique to infinitely non-unique. The mathematics that describes gas dynamics is, in two dimensions, insufficiently constrained by the physics it's meant to capture. Additional selection principles — beyond the standard entropy condition — are needed to pick the physical solution, and the theory doesn't yet know what those principles are.