AI 中文总结
该研究针对s∈[1/2,1)的分数阶漂移扩散方程，建立了粘性解的梯度正则性与位势估计，推导了相关稳定性结果及全空间核的极限性质，明确了漂移到边界时的障碍。
AI 中文摘要
针对s∈[1/2,1)、局部Hölder连续的b和f对应的(-Δ)^s u + b·∇u = f的有界粘性解，我们建立了尺度不变的内部C^{1,α}估计：临界情形采用Silvestre的抛物型定理，次临界情形采用Schauder估计与插值。将该粘性估计应用于带漂移的Green截面，对临界阶以上的有限Radon数据，我们得到阶为2s和2s-1的精确解与梯度位势，以及弱*到强的局部W^{1,1}稳定性，故Green位势SOLA与近似无关。对归一化全空间核，我们确定了s↑1时的经典二阶极限，包括二维的对数核；在临界阶，全空间梯度成为零阶奇异积分。对漂移紧支的零外问题，我们还证明u/d^s∈C^{s-ε}(Ω̄)，并确定漂移到达边界时的障碍。
英文摘要
We establish scale-invariant interior $C^{1,α}$ estimates for bounded viscosity solutions of $(-Δ)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally Hölder $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this viscosity estimate to drifted Green sections, for finite Radon data above the critical order we obtain sharp solution and gradient potentials of orders $2s$ and $2s-1$, together with weak-*--to--strong local $W^{1,1}$ stability; hence the Green-potential SOLA is approximation-independent. For the normalized whole-space kernels, we identify the classical second-order limits as $s\uparrow1$, including the logarithmic kernel in dimension two; at the critical order, the whole-space gradient becomes a zero-order singular integral. For zero-exterior problems with compactly supported drift, we also prove $u/d^s\in C^{s-\varepsilon}(\overlineΩ)$ and identify the obstruction when the drift reaches the boundary.
