Genuinely Nonlinear Scalar Conservation Laws

Abstract

Despite its apparent simplicity, the genuinely nonlinear scalar conservation law in one space dimension possesses a surprisingly rich theory, which deserves attention not only for its intrinsic interest but also because it provides valuable insight in the behavior of systems. The discussion here will employ the theory of generalized characteristics developed in Chapter X. From the standpoint of this approach, the special feature of genuinely nonlinear scalar conservation laws is that the extremal backward generalized characteristics are essentially classical characteristics, that is, straight lines along which the solution is constant. This property induces such a heavy constraint that one is able to derive very precise information on regularity and large time behavior of solutions. Solutions are (classically) continuous at points of approximate continuity and locally Lipschitz continuous in the interior of the set of points of continuity. Points of approximate jump discontinuity lie on classical shocks. The remaining, irregular, points are at most countable and are formed by the collision of shocks and/or the focusing of compression waves. Generically, solutions with smooth initial data are piecewise smooth.