Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Regularity and large time behaviour of solutions of a conservation law without convexity

C. M. Dafermosa1

a1 Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A.

Synopsis

Using the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.

We show that when the initial data are C then, generically, the solution is C except: (a) on a finite set of C arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.

We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.

(Received May 17 1984)