Finite-time gradient blow-up and shock formation in Israel-Stewart theory: Bulk, shear, and diffusion regimes

We present a demonstration of finite-time gradient blow-ups in Israel-Stewart (IS) theories with 1+1D plane symmetry, mathematically showing the existence of smooth initial data that can evolve into shocks across three regimes: pure bulk viscosity, shear viscosity, and diffusion. Through numerical simulations of bulk-viscous fluids, we verify that these shocks satisfy Rankine-Hugoniot conditions, exhibit characteristic velocity crossing (the Mach number obeys M_{u}>1>M_{d}), and maintain thermodynamic consistency, required for physical shocks. Our results reveal a crucial early-time dynamical phase where nonlinear effects dominate viscous damping, resolving the apparent impossibility of IS-type theories predicting shock formation. While restricted to simplified 1+1D systems with separate viscous effects, this work establishes foundational insights for shock formation in relativistic viscous hydrodynamics, highlighting critical challenges for extending to 3+1D systems or to a full IS theory where multiple nonlinear modes interact. The findings emphasize that both initial data structure and numerical methodology require careful consideration when studying shocks in relativistic viscous fluids.