Constructing unbounded discontinuous solutions of scalar conservation laws using the Legendre transform

A first-order quasilinear equation with an odd flux function that has a single point of inflexion at zero is studied. A method for constructing sign-alternating discontinuous entropy solutions of this equation, based on the Legendre transform, is proposed. Bibliography: 18 titles. © 2021 Russian Academy of Sciences (DoM) and London Mathematical Society.