Today nonlocal differential equations have become an efficient tool for modelling complex phenomena in fluid mechanics, life science, hydrology, geophysics and biology system and their mathematical aspects have been extensively investigated.
Semigroup theory is one of the powerful tools for studying the partial differential equations.
Huang, Wang, Zheng and Duan [J. Differ. Equ., 265(2018) 4181-4212] obtained a resolvent estimate which shows that the fractional Laplacian generates an analytic semigroup in Lp(Rn). Maekawa and Miura [Adv. Math., 247 (2013) 123191] proved that-((-Δ)β)+b(x)•▽ generates a strongly continuous semigroup in Lp(Rn) under some conditions on b(x). Wang, Ma and Duan[J. Math. Anal. Appl., 481(2020) 123480] verified that when b is constant,-((-Δ)β)+b•▽ generates an analytic semigroup if 1/2≤β<1and a Gevrey semigroup of order δ with δ>1/(2β) if 0<β<1/2 in Lp(Rn). Moreover, they showed that the above mentioned result on Gevrey regularity for0<β<1/2 is optimal.
We investigated the regularity of operator semigroups generated by the gradient perturbations of important nonlocal operators such as Riesz-Feller operator and exponentially tempered fractional Laplacian. By using the Fourier multiplier theory to obtain resolvent estimates for Riesz-Feller operator and exponentially tempered fractional Laplacian, we proved that the result on Gevrey regularity in [J. Math. Anal. Appl., 481(2020) 123480] still holds in our cases.
Our research results were published in "Complex Analysis and Operator Theory, 17 (2023) 49" and "Journal of Mathematical Analysis and Applications" under the title of "Gevrey type regularity of the Riesz-Feller operator perturbed by gradient in Lp(Rn)"(https://doi.org/10.1007/s11785-023-01354-8) and "Regularity of semigroups for exponentially tempered stable processes with drift"(https://doi.org/10.1016/j.jmaa.2023.127247).
