LINEAR CAUCHY PROBLEM WITH VARIABLE COEFFICIENTS

Abstract

Abstract This chapter emphasizes energy estimates, that is, a priori estimates without loss of derivatives in Sobolev spaces. The appropriate tools are functional symmetrizers, obtained merely as Fourier multipliers for Friedrichs symmetrizable operators, and constructed by means of symbolic symmetrizers for more general operators. In particular, symbolic symmetrizers are shown to exist for constantly hyperbolic operators. Subsequently, well-posedness is proved in Sobolev spaces Hs, for any s in the case of infinitely smooth coefficients, and also for s large enough in the case of Hs coefficients. The existence of solutions relies on the Hahn-Banach and Riesz theorems, using energy estimates for the adjoint backward Cauchy problem. Uniqueness follows from energy estimates for the direct Cauchy problem. Regularity is shown by means of the weak equals strong argument, which is shown as the weak solutions obtained by the duality method are necessarily strong solutions, that is, suitable limits of infinitely smooth solutions of regularized problems.