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

Abstract

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 $w_t+(-\Delta {)}^{s}w=h;w(x,t)=0\text{ in }({\mathbb{R}}^N\setminus \Omega )\times (0,T);w(x,0)=w_0(x)\text{in}\Omega$, where $\Omega$ 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.

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