Two by Two Systems with Two Independent Variables

Abstract

In this chapter we study the C ^∞ well-posedness of the Cauchy problem for 2 × 2 systems with two independent variables with real analytic coefficients. For such a system L the characteristic set is given by zeros of some nonnegative real analytic function. We define pseudo-characteristic curves for L as the real part of the zeros of nonnegative functions associated to the system and we give a necessary and sufficient condition for the Cauchy problem for L to be C ^∞ well posed in terms of pseudo-characteristic curves and Newton polygons. In particular we can characterize strongly hyperbolic 2 × 2 systems with two independent variables. This gives another proof of the strong hyperbolicity of the 2 × 2 system discussed in Sect. 1.3. By checking this necessary and sufficient condition we provide many instructive examples. For instance, we see that there are examples which are strictly hyperbolic apart from the initial line with polynomial coefficients such that the Cauchy problem is not C ^∞ well posed for any lower order term.