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This paper is devoted to strictly hyperbolic systems and equations withnon-smooth coefficients. Below a certain level of smoothness, distributionalsolutions may fail to exist. We construct generalised solutions in theColombeau algebra of generalised functions. Extending earlier results onsymmetric hyperbolic systems, we introduce generalised strict hyperbolicity,construct symmetrisers, prove an appropriate G\r{a}rding inequality andestablish existence, uniqueness and regularity of generalised solutions. Underadditional regularity assumptions on the coefficients, when a classicalsolution of the Cauchy problem (or of a transmission problem in the piecewiseregular case) exists, the generalised solution is shown to be associated withthe classical solution (or the piecewise classical solution satisfying theappropriate transmission conditions).

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients