Commentary on [77], [85], [92], and [94]

Abstract

One major line of interest of F. John over the last eight to nine years has been the study of long-time behavior of classical solutions to quasilinear hyperbolic equations. His motivation was to understand the connections between linear and nonlinear elasticity, however his results have already stimulated a great deal of research in the broad area of nonlinear hyperbolic equations. In his first such paper he considers quasilinear strictly hyperbolic equations in one space dimension. In connection with the equations of one-dimensional gasdynamics, it was well known that initially smooth solutions of such equations generally develop singularities in their first derivatives in a finite time. In gasdynamics this spontaneous appearance of singularities corresponds to formation of shock waves, a fact which is well observed in physical reality. The meaning of such singularities is, however, less clear for the equations of elasticity and their presence subject to possible controversy. In [77] F. John shows that the same type of singularities as in gasdynamics are generally present for the 3x3 system of nonlinear wave equations satisfied by plane- wave solutions to nonlinear elasticity. Reducing the 3x3 second-order system to a first-order one, he proves a general result concerning the formation of singularities for strictly hyperbolic “genuinely nonlinear” (in the sense of Lax) first-order systems. Previous general results of this type were known only for genuinely nonlinear 2x2 systems ([C. 1]) and were dependent on the possibility of diagonalizing such systems with the help of Riemann invariants. The general systems dealt with by F. John are not diagonalizable, and thus the mechanism of formation of singularities is much more complicated.