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Energy Solutions and Harnack Estimates for Nonlocal Drift–Diffusion

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Contributed by Zhenyu Sun ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

We consider nonlocal drift–diffusion equations of order 2s∈[1,2)2s\in[1,2) with bounded divergence-free drift. The symmetric measurable jump kernel satisfies local coercivity, a global upper form bound, and a scale-invariant bound on the intensity of long jumps. The transport form is bounded and skew-symmetric on H1/2×H1/2H^{1/2}\times H^{1/2}. In the fractional energy space, distributional and variational solutions coincide and satisfy nonlinear testing identities. Viscous approximations converge strongly in CtLx2∩Lt2HxsC_tL^2_x\cap L^2_tH^s_x to the unique solution of the Cauchy problem. For sources bounded in the cylinder, supersolutions nonnegative there satisfy a weak Harnack inequality involving the time integral of the negative exterior tail. Under this source bound, locally bounded solutions whose exterior tail belongs to LqL^q in time for some q>1q>1 are locally Hölder continuous. This also gives Hölder continuity at positive times for homogeneous global solutions with L2L^2 data. Under upper jumping smoothness, control of the positive tail on an earlier time interval yields the full parabolic Harnack inequality for solutions nonnegative on the full spatial slab. A kernel supported on alternating dyadic annuli satisfies all the kernel assumptions. The local estimates depend on the drift through R2s−1∥b∥∞R^{2s-1}\|b\|_\infty.

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Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentNov 03, 2026