Sharp L1 stability estimates for hyperbolic conservation laws

In this paper, we introduce a generalization of Liu-Yang's weighted norm to linear and to nonlinear hyperbolic equations. Extending a result by Hu and LeFloch for piecewise constant solutions, we establish sharp L1 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 L1 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 L1 estimate directly from the equation.