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von Neumann: I would like to make some remarks about the general hydrodynamical discussion on motions in one dimension, following Riemann’s theory, which was expounded by Dr. Burgers and subjected to a critical analysis by Dr. McVittie. The question as to whether a solution which one has found by mathematical reasoning really occurs in nature and whether the existence of several solutions with certain good or bad features can be excluded beforehand, is a quite difficult and ambiguous one. This subject has been considered in the classical literature as well as in the more recent literature, on widely varying levels of rigor and of its opposite. In summa, it is quite difficult ever to be sure of anything in this domain. Mathematically, one is in a continuous state of uncertainty, because the usual theorems of existence and uniqueness of a solution, that one would like to have, have never been demonstrated and are probably not true in their obvious forms. To this day, the only thing of any degree of generality that we possess is the classical discussion by Riemann, and this very strictly in one dimension and very strictly in the isentropic case. In this case at least, Riemann proved that there are no discontinuities. He also gave the exact conditions under which there can be a solution at all and he proved that in those cases there is only one. So he proved that the number of solutions is either zero or one. He also showed that it is zero in general, i.e., unless certain (infinitely many) very stringent conditions are satisfied. Thus, unless the initial state of the gas fulfills some very particular conditions, the (continuous) solution will cease to exist after some definite finite time. Riemann also inferred, essentially by physical insight, what happens when the continuous solution ceases to exist. He made it very plausible that a discontinuity of a certain type, a “shock wave”, develops. This was subsequently independently rediscovered, and further developed, by Hugoniot. It is also true that in the entire literature up to 1910, i.e., up to the time of the work of Rayleigh and G. I. Taylor, there was a considerable confusion

Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations