Rosella Sampalmieri

Global existence to the Cauchy problem for hyperbolic conservation laws with an isolated umbilic point

In this paper the existence of global weak solutions for a 2 × 2 system of non-strictly hyperbolic non-linear conservation laws is established for data in L∞. The result is proven by means of viscous approximation and application of the compensated compactness method. The presence of a degeneracy in the hyperbolicity of the system requires a careful analysis of the entropy functions, whose regularity is necessary to obtain an existence result. For this purpose we combine the classical techniques referring to a singular EulerPoisson-Darboux equation with the compensated compactness method.

Light rays having extreme points with the same spatial coordinates

Abstract In this paper we deal with lightlike geodesics in Lorentzian manifolds which are closed with respect to the spatial coordinates. We consider the case where the initial point is fixed or moves on a level of a given time function. The case where the initial point is free is also considered, under a periodicity condition on the metric.