Decay Estimates for Hyperbolic Balance Laws

Abstract

This work is concerned with time-decay properties of small-amplitude global smooth solutions to the initial value problem for hyperbolic systems of balance laws admitting an entropy and satisfying the stability condition. By using energy methods in both the physical space and the Fourier space, we obtain the optimal decay estimates of solutions and their derivatives in the $L^2$-norm up to order $s-1$, provided that the initial data are in $H^s$. A key ingredient in our analysis is a time-weighted energy estimate, leading to a decay estimate for multi-dimensional problems without assuming the $L^1$ property on initial data.