We discuss the global existence of weak (entropy) solutions for two distinct classes of hyperbolic systems by constructing novel viscous regularizations. In the first part of the talk, we focus on a class of $2 \times 2$ Keyfitz-Kranzer systems arising in thin film flows. Because the standard diagonal viscous approximation fails to yield the appropriate mathematical structure, we instead construct a novel viscous regularization inspired by lubrication equations. This approach provides the critical compactness estimates necessary to establish the existence of weak entropy solutions via the vanishing diffusion limit.

In the second part of the talk, we generalize these results to $n \times n$ systems of a broader Keyfitz-Kranzer class covering several important physical models. We introduce a new artificial viscous regularization aligned with the nonlinear field, which yields a parabolic equation for the Riemann invariant associated with the nonlinear field, coupled with transport equations for the remaining Riemann invariants. This structure drives the compactness argument and yields global existence of weak solutions for the inviscid system. Based on joint works with Christian Rohde (University of Stuttgart, Stuttgart, Germany), Phillipp Offner (TU Clausthal) and Lilu Sahu (IIT Kharagpur).