On Schauder estimates for nonlocal and mixed local-nonlocal viscous Hamilton$\unicode{x2013}$Jacobi equations

TL;DR

This paper establishes optimal Schauder regularity estimates and well-posedness for viscous Hamilton–Jacobi equations involving nonlocal and mixed local-nonlocal diffusions, covering various operators and initial data conditions.

Contribution

It provides the first comprehensive Schauder estimates for a broad class of nonlocal and mixed operators in viscous Hamilton–Jacobi equations, including explicit blow-up rates.

Findings

Optimal regularity in Hölder spaces achieved

Short and long time existence of solutions proven

Explicit blow-up rates for Hölder norms as time approaches zero

Abstract

We prove spatial 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 results 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 give results for the 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.