L 2 Formulation of Multidimensional Scalar Conservation Laws

We show that Kruzhkov's theory of entropy solutions to multidimensional scalar conservation laws (Kruzhkov in Mat Sb (N.S.), 81(123), 228-255, 1970) can be entirely recast in L (2) and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Brenier (in C R Acad Sci Paris Ser I Math, 292, 563-566, 1981; in J Diff Equ, 50, 375-390, 1983; in SIAM J Numer Anal, 21, 1013-1037; in Methods Appl Anal, 11, 515-532, 2004), Giga and Miyakawa (in Duke Math J, 50, 505-515, 1983), and Tsai et al. (in Math Comp, 72, 159-181, 2003).