Identification of arbitrary distributional solutions with viscosity solutions for Hölder drifts

Establish whether an arbitrary distributional solution of the fractional drift–diffusion equation $(-\Delta)^s u+b\cdot\nabla u=f$ with merely $C^\gamma$ drift can be identified with a viscosity solution without assuming that it arises from the specified smooth approximation or satisfies an equivalent comparison property.

Background

The paper proves equivalence between local energy weak solutions and viscosity solutions under a divergence-free, locally Lipschitz drift. It then explains that, when the drift is only Hölder continuous, the bilinear drift form need not automatically extend to the fractional energy space, so the energy-based argument no longer applies to arbitrary distributional solutions.

The authors state that the viscosity result itself remains valid for merely Hölder drifts, but they do not establish that every distributional solution belongs to the viscosity framework. The unresolved issue is whether such an identification can be proved without either a smooth-approximation construction or an additional comparison property.