Global regularity results for the fractional heat equation and application to a class of non-linear KPZ problems

In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation $w_t+(-\mathrm{\Delta }{)}^{s}w=h;w(x,t)=0\text{ in }({\mathbb{R}}^N\setminus \mathrm{\Omega })\times (0,T);w(x,0)=w_0(x)\text{in}\mathrm{\Omega }$, where $\mathrm{\Omega }$ is an open bounded subset of ${\mathbb{R}}^N$. The proof is based on a new pointwise estimate on the fractional gradient of the corresponding kernel. Moreover, we establish the compactness of $(w_0,h)\mapsto w$. As a majeur application, in the second part , we establish existence and regularity of solutions to a class of Kardar–Parisi–Zhang equations with fractional diffusion and a nonlocal gradient term. Additionally, several auxiliary results of independent interest are obtained.