Sup - norm Estimates in Glimm ’ s Method *

We give a proof that Total Variation {a,,(.)} < 1 can be replaced by Sup{u,(.)} e 1 in Glimm's method whenever a coordinate system of Riemann invariants is present. The argument is somewhat simpler but in the same spirit as that given by Glimm in his celebrated paper of 1965. 0 1990 Academic Press, Inc. We consider the Cauchy problem u, + F(u), = 0, (1) 4-G 0) = u,(x), (2) where (1) denotes a strictly hyperbolic system of two conservation laws, = 1, 2 be the eigenvalues and corresponding eigenvector fields associated with the matrix VF, A, < &. Assume that @ is a neighborhood of a state U in u-space in which each characteristic field is either genuinely nonlinear (VA,. R, > 0) or else linearly degenerate (VA,. R, =O), and such that n,(u) < n,(u) for all U, OE@[~]. Without loss of generality, assume ii = 0. In this note we give a simplified proof of the following sup-norm estimate which is contained in the results of Glimm [2] and which is required for the proof in [12]. (A stronger result also follows from the analysis in [3] which, however, involves the theory of approximate characteristics and is much more technical.) We comment that a corresponding sup-norm estimate for more than two equations is all that is required in order to extend the results in [12, 131 to systems of more than two equations. We anticipate that the ideas here will help in the proof of such an estimate. Let u(x, t) denote a weak solution of (1), (2) which is a limit of approximate solutions generated by the random choice method of Glimm.