Inverse problems for scalar conservation laws

The weak solution to the initial value problem for scalar conservation laws may develop a discontinuity called a shock. In this paper, we consider an inverse problem to determine the flux function entering the scalar conservation law by observing the shock developed by a single initial datum. We prove that the flux function on an interval can be uniquely determined by the shock. We also prove that this interval can be taken arbitrarily large by choosing an appropriate sequence of initial data. We then show that the derivatives of the flux function, and those of initial data, can be reconstructed in an explicit way from the coefficients of asymptotics of the shock wave.