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Zeitschrift für Analysis und ihre Anwendungen


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Volume 28, Issue 1, 2009, pp. 1–33
DOI: 10.4171/ZAA/1369

Published online: 2009-03-31

Decay Estimates for Hyperbolic Balance Laws

Shuichi Kawashima[1] and Wen-An Yong[2]

(1) Kyushu University, Fukuoka, Japan
(2) Tsinghua University, Beijing, China

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.

Keywords: Hyperbolic balance laws, entropy, stability condition, global smooth solutions, decay estimates

Kawashima Shuichi, Yong Wen-An: Decay Estimates for Hyperbolic Balance Laws. Z. Anal. Anwend. 28 (2009), 1-33. doi: 10.4171/ZAA/1369