A representation formula of viscosity solutions to weakly coupled systems of Hamilton-Jacobi equations with applications to regularizing effect

TL;DR

This paper introduces a dynamical representation formula for viscosity solutions of weakly coupled Hamilton-Jacobi systems, leading to new regularity results and extending Lions' regularizing effect to these systems.

Contribution

It provides a novel representation formula for viscosity solutions of coupled systems, enabling the extension of regularity results to more complex equations.

Findings

Derived a dynamical representation formula for solutions

Extended Lions' regularizing effect to coupled systems

Established new regularity results for viscosity solutions

Abstract

Based on a fixed point argument, we give a {\it dynamical representation} of the viscosity solution to Cauchy problem of certain weakly coupled systems of Hamilton-Jacobi equations with continuous initial datum. Using this formula, we obtain some regularity results related to the viscosity solution, including a partial extension of Lions' regularizing effect \cite{L} to the case of weakly coupled systems.