math.AP — Analysis of PDEs

Energy Solutions and Harnack Estimates for Nonlocal Drift–Diffusion

Contributed by Zhenyu Sun

We consider nonlocal drift–diffusion equations of order $2s\in[1,2)$ 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 $H^{1/2}\times H^{1/2}$. In the fractional energy space, distributional and variational solutions coincide and satisfy nonlinear testing identities. Viscous approximations converge strongly in $C_tL^2_x\cap L^2_tH^s_x$ 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 $L^q$ in time for some $q>1$ are locally Hölder continuous. This also gives Hölder continuity at positive times for homogeneous global solutions with $L^2$ 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 $R^{2s-1}\|b\|_\infty$.