On the differentiability of solutions of symmetric hyperbolic systems
Proceedings of the American Mathematical Society
This paper is concerned with the Differentiability Theorem for linear symmetric hyperbolic systems of partial differential equations of the first order. In [l] Friedrichs derives the existence and uniqueness of the strong solution of the Cauchy problem for these systems by using energy inequalities and orthogonal projections. His main tool is the integral mollifier. However, to show that the solution possesses square integrable (strong) derivatives, provided the data determining it is
... ing it is sufficiently smooth, he uses the method of approximations by finite differences. In a different approach, Lax  introduces spaces with norms of negative order, such that his solution is already equipped with the necessary square integrable derivatives. We shall present here a direct proof of the strong differentiability of the solution by applying the existence theorem to the over-determined system resulting from differentiating the given system with respect to all the space variables. The problem of the strong differentiability of the solution is of particular importance in light of Sobolev's lemma  , which states that a function possesses continuous derivatives in an appropriate domain, provided it possesses there a sufficient number of square integrable derivatives. Let u= (ui, • • • , un) denote a vector function of n elements in the m+1 independent variables (t, x) = (t,xi, • • • ,xm). Ai, t = l, • • • ,m, are symmetric «X« matrices with sufficiently continuously differentiable elements. B is an nXn matrix with sufficiently continuously differentiable elements. The domain R is the infinite slab between Received by the editors September 17, 1962.