SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension

We prove the SBV regularity of the characteristic speed of the scalar hyperbolic conservation law and SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux function for general hyperbolic systems of conservation laws. More precisely, for the equation ut + f(u)x = 0, u : R + × R → Ω ⊂ RN , we only assume that the flux f is a C2 function in the scalar case (N = 1) and Jacobian matrix Df has distinct real eigenvalues in the system case (N ≥ 2). Using a modification of the main decay estimate in [8] and the localization method applied in [17], we show that for the scalar equation f0(u) belongs to the SBV space, and for system of conservation laws the i-th component of Dxλi(u) has no Cantor part, where λi is the i-th eigenvalue of the matrix Df.