We investigate regularity and a priori estimates for Fokker–Planck and Hamilton–Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order \(s\in (1/2,1)\). As for Fokker–Planck equations, we establish integrability estimates under a fractional version of the Aronson–Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans’ nonlinear adjoint method we prove new integral, sup-norm and Hölder estimates for weak and strong solutions to fractional Hamilton–Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal \(L^q\)-regularity for fractional Hamilton–Jacobi equations.

我们研究了具有无界成分的 Fokker-Planck 和 Hamilton-Jacobi 方程的规律性和先验估计，该方程由阶数\(s\in (1/2,1)\)的分数拉普拉斯算子驱动. 对于 Fokker-Planck 方程，当漂移相对于解本身具有低 Lebesgue 可积性时，我们在速度场和 Bessel 正则性的 Aronson-Serrin 插值条件的分数版本下建立可积性估计。使用这些估计，通过 Evans 的非线性伴随方法，我们证明了具有无界右侧和梯度多项式增长的分数阶 Hamilton-Jacobi 方程的弱解和强解的新积分、超范数和 Hölder 估计。最后，通过后面的这些结果，利用线性非局部偏微分方程和分数 Gagliardo-Nirenberg 不等式的 Calderón-Zygmund 型正则性，我们推导出分数 Hamilton-Jacobi 方程的最优\(L^q\) -正则性。