Conormal and Piecewise Smooth Solutions

In this paper, we show first that if a solution u of the equation i*2(l, x, u, Du, D)u = f{t, x,u, Du), where Pi{t, x,u, Du, D) is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface S of Pi in the past and S is smooth in the past, then S is smooth and u is conormal with respect to S for all time. Second, let So arjd Si be characteristic hypersurfaces of Pi which intersect transversally and let T = So nS] . If So and Si are smooth in the past and u is conormal with repect to {So , S]} in the past, then T is smooth, and u is conormal with respect to {So , S[} locally in time outside of T, even though S0 and S) are no longer necessarily smooth across T. Finally, we show that if «(0, x) and 3,w(0, x) are in an appropriate Sobolev space and are piecewise smooth outside of T, then u is piecewise smooth locally in time outside of So uS, .