Gradient regularity for mixed local-nonlocal quasilinear parabolic equations

Abstract

In this paper, we prove local H\"older continuity for the spatial gradient of weak solutions to $$u_t - \text{div} (|\nabla u|^{p-2}\nabla u) + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+ps}} \ dy = 0.$$ It is easy to see that parabolic quasilinear equations are not scaling invariant and this led to the development of the method of intrinsic scaling by E.DiBenedetto, E.DiBenedetto-Y.Z.Chen, J.Kinnunen-J.Lewis and A.Friedman-E.DiBenedetto. In a very recent paper, C.de Filippis-G.Mingione proved gradient H\"older continuity for mixed local-nonlocal quasilinear elliptic equations and in this paper, we extend this result to the parabolic case. Since we only expect regularity for $\nabla_x u$ in the parabolic setting, it is not clear how to extend the elliptic proof to the parabolic case. In order to overcome this difficulty, we instead follow the ideas developed by T.Kuusi-G.Mingione combined with the novel tail estimates of C.deFilippis-G.Mingione. An advantage of our approach is that we can obtain both $C^{1,\alpha}_x$ regularity as well as $C^{0,1} _x$ potential estimates in one go. Moreover, we do not need to make use of any form of Caccioppoli inequality and instead, the regularity is obtained only through a suitable difference estimate.