Uniqueness of entropy solutions in the general non‑radial setting

Establish whether entropy solutions to the scalar conservation law on the d‑dimensional torus, given by ∂_t u − div(u^m ∇g ∗ u) = 0 with g the Green’s kernel satisfying −Δg = δ_0 − 1, are unique for general initial data and any m > 0 beyond the already known one‑dimensional and radial cases. In particular, determine uniqueness within the class of entropy solutions without imposing radial symmetry.

Background

The paper studies a nonlocal conservation law modeling repulsive interactions with nonlinear mobility on the torus. While existence of entropy solutions is established via a vanishing viscosity method and a weak–strong uniqueness principle is proved, general uniqueness for entropy solutions is not known. Uniqueness has been shown only in special cases—one spatial dimension and radial symmetry—by exploiting the Hamilton–Jacobi structure available in those settings.

Addressing uniqueness in the non‑radial, higher‑dimensional case is central for well‑posedness theory and for the robustness of qualitative properties such as front dynamics and asymptotic behavior. The difficulty is compounded by potential shock formation and the absence of a comparison principle.