# Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative

## TL;DR

This paper investigates the existence, uniqueness, and regularity of classical solutions to viscous Hamilton-Jacobi equations involving Caputo time-fractional derivatives, using advanced mathematical techniques.

## Contribution

It introduces new methods to establish regularity and uniqueness for fractional Hamilton-Jacobi equations, combining gradient bounds and sharp Sobolev and Hölder estimates.

## Key findings

- Proved existence and uniqueness of solutions.
- Established regularity properties in Sobolev and Hölder spaces.
- Developed a gradient bound using nonlinear adjoint methods.

## Abstract

We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and H\"older spaces for the corresponding linear problem.