Sharp $L^1$ Stability Estimates for Hyperbolic Conservation Laws

P. Goatin and P.G. LeFloch

Abstract: In this paper, we introduce a generalization of Liu--Yang's weighted norm to linear and to nonlinear hyperbolic equations. Following an approach due to the second author for piecewise constant solutions, we establish sharp $L^1$ continuous dependence estimates for general solutions of bounded variation. Two different strategies are successfully investigated. On one hand, we justify passing to the limit in an $L^1$ estimate valid for piecewise constant wave-front tracking approximations. On the other hand, we use the technique of generalized characteristics and, following closely an approach by Dafermos, we derive the sharp $L^1$ estimate directly from the equation.

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