Well-posedness of Hamilton-Jacobi equations with Caputo's time-fractional derivative

TL;DR

This paper establishes the well-posedness of Hamilton-Jacobi equations involving Caputo's time-fractional derivatives, introducing viscosity solutions, and analyzing stability, regularity, and applications to linear transport equations.

Contribution

It introduces a notion of viscosity solutions for fractional Hamilton-Jacobi equations and proves their existence, uniqueness, stability, and regularity under periodic boundary conditions.

Findings

Unique existence of solutions is proven.

Comparison principle and Perron's method are established.

Results apply to linear transport equations with variable coefficients.

Abstract

A Hamilton-Jacobi equation with Caputo's time-fractional derivative of order less than one is considered. The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions. For this purpose comparison principle as well as Perron's method is established. Stability with respect to the order of derivative as well as the standard one is studied. Regularity of a solution is also discussed. Our results in particular apply to a linear transport equation with time-fractional derivatives with variable coefficients.