ABSTRACT
The great advantage of equation (379) is that it is in the normal form, so it obeys the Cauchy-Kovalevskaya theorem [385, p. 387]. Hence, for any initial analytic data u(x, 0), ut(x, 0), utt(x, 0), uttt(x, 0), and utttt(x, 0), there exists a unique local in time analytic solution u(x, t). Thus, (379) generates a local semigroup of analytic solutions, and this makes it easier to deal with smooth δ-deformations that are chosen to be analytic. This defines a special analytic δ-entropy test for shock/rarefaction waves. On the other hand, such nonlinear PDEs can admit other (say, weak) solutions that are not analytic. Actually, Proposition 8.88 shows that the shock S−(x) is a δ-entropy solution of (379), which is obtained by a finite-time blow-up, as t → 0−, from the analytic similarity solution (497).
