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

Abstract

We present the first 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 (Mach number obeys ${\mathcal{M}}_u>1>{\mathcal{M}}_d$), and maintain thermodynamic consistency, required for physical shocks. Our results reveal a crucial early-time dynamical phase -- previously unexplored in steady-state analyses -- 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.

25 pages, 8 figures. Some text added. Peer reviewed version

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