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

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