A Schauder regularity theory for nonlocal and mixed local-nonlocal viscous Hamilton$\unicode{x2013}$Jacobi equations

Abstract

We prove space-time Schauder estimates $\unicode{x2013}$ optimal regularity estimates in H\"older spaces $\unicode{x2013}$ and well-posedness results for mild and classical solutions of viscous Hamilton$\unicode{x2013}$Jacobi equations with subcritical nonlocal and mixed local-nonlocal diffusions in $\mathbb{R}^d$. Our spatial Schauder estimates hold under mild assumptions on the nonlocal/mixed operators and Hamiltonians. The Laplacian, fractional Laplacians, nonsymmetric, spectrally one-sided, and strongly anisotropic integral operators, as well as sums of such operators are covered. We observe an interplay between the regularity of the initial data and the growth of the Hamiltonian in the gradient, and develop a spatial Schauder theory for two canonical cases: (i) Lipschitz initial data and general Hamiltonians that are H\"older in space and merely locally Lipschitz in the gradient, and (ii) H\"older initial data and Hamiltonians that are H\"older in space and locally Lipschitz with power growth in the gradient. We compute explicit blow-up rates for $C^1$ and higher order H\"older norms as $t\to 0$. The results include short and long time existence of mild solutions, optimal regularity in H\"older spaces and corresponding Schauder a priori estimates, and that spatially smooth mild solutions are regular in time and pointwise classical solutions. Under further assumptions on the diffusion operator, we then prove time and space-time Schauder regularity estimates in optimal H\"older spaces which respect the natural fractional parabolic scaling. These results generalize classical linear local and fractional Schauder estimates to our non-linear fractional, possibly anisotropic and nonsymmetric setting.