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

Fábio S. Bemfica

TL;DR

This work addresses whether shocks can form in causal relativistic viscous fluids described by Israel-Stewart theory. It analyzes strictly hyperbolic, first-order IS equations in $1+1$ dimensions across three dissipation channels (bulk, shear, diffusion), combining Barlin-style gradient blow-up theory with high-resolution shock-capturing numerics. The main finding is the existence of smooth initial data that develop finite-time gradient blow-ups, leading to true shocks while the dynamical variables remain bounded, with shocks satisfying Rankine-Hugoniot conditions and exhibiting Mach-number crossing in the shock frame. The results illuminate an early-time nonlinear regime where dissipative effects do not yet erase steep gradients, complementing classic steady-state analyses (Olson-Hiscock, Geroch-Lindblom) and outlining a path toward understanding shocks in the full, higher-dimensional IS framework and in realistic heavy-ion collision dynamics. Limitations include the 1+1D, separately treated viscous sectors, and the need for extending to $3+1$D with coupled viscous mechanisms and more general equations of state. The findings provide foundational insight into shock formation in causal relativistic hydrodynamics and bear potential relevance for pre-equilibrium dynamics in high-energy nuclear collisions.

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.

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

TL;DR

This work addresses whether shocks can form in causal relativistic viscous fluids described by Israel-Stewart theory. It analyzes strictly hyperbolic, first-order IS equations in dimensions across three dissipation channels (bulk, shear, diffusion), combining Barlin-style gradient blow-up theory with high-resolution shock-capturing numerics. The main finding is the existence of smooth initial data that develop finite-time gradient blow-ups, leading to true shocks while the dynamical variables remain bounded, with shocks satisfying Rankine-Hugoniot conditions and exhibiting Mach-number crossing in the shock frame. The results illuminate an early-time nonlinear regime where dissipative effects do not yet erase steep gradients, complementing classic steady-state analyses (Olson-Hiscock, Geroch-Lindblom) and outlining a path toward understanding shocks in the full, higher-dimensional IS framework and in realistic heavy-ion collision dynamics. Limitations include the 1+1D, separately treated viscous sectors, and the need for extending to D with coupled viscous mechanisms and more general equations of state. The findings provide foundational insight into shock formation in causal relativistic hydrodynamics and bear potential relevance for pre-equilibrium dynamics in high-energy nuclear collisions.

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 ), 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.

Paper Structure

This paper contains 13 sections, 4 theorems, 95 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Early-Time Nonlinear Dominance vs. Late-Time Viscous Dominance
  3. Mathematical framework
  4. Approximate Estimates of Gradient Blow-ups
  5. Barotropic Fluid with Bulk Viscosity
  6. Barotropic fluid with Shear Viscosity
  7. Israel-Stewart Theory with Diffusion
  8. Shock Formation in Bulk Viscous Equations - Numerical Study
  9. Oppositely-Propagating Shock Formation (Case 1)
  10. Rarefaction-Shock Wave Configuration (Case 2)
  11. Conclusions
  12. Numerical scheme
  13. Bjorken Flow

Key Result

Theorem 1

Under Assumptions Assumption_T1, Assumption_T2, and Assumption_T3, there exist smooth initial data $\overset{\circ}{\psi} \colon \mathbb{R} \to \Psi$ such that the unique $C^1$-solution of EOM-Matrix2 with $\psi(0,x) = \overset{\circ}{\psi}(x)$ exists only for finite time, while $\psi(t,x)$ remains

Figures (8)

  • Figure 1: Time evolution at $t=4\,\text{GeV}^{-1}$ showing profiles of (a) velocity $v$, (b) energy density $\varepsilon$, and (c) bulk viscosity $\Pi$ with WENO-Z and MUSCL numerical schemes for comparison. We used $\Delta x=1/400$. Figure (d) contains the velocity profile at $t=1,2,3,4\,\text{GeV}^{-1}$, solved with WENO-Z. Mode features are labeled: (I) shock ($\lambda_-$ mode), (II) linear wave, (III) shock ($\lambda_+$ mode).
  • Figure 2: Evolution at $t=4\,\text{GeV}^{-1}$ showing: (a) velocity $v$, (b) energy density $\varepsilon$, (c) bulk viscosity $\Pi$ (WENO-Z and MUSCL), and (d) $\varepsilon$ evolution at $t=1,2,3,4\,\text{GeV}^{-1}$ (WENO-Z). We used grid size $\Delta x =1/400$. Features labeled: (S) shock, (R) rarefaction, (L) linear wave.
  • Figure 3: Convergence factors $\tilde{p}_a$ for (a) WENO-Z and (b) MUSCL schemes, showing expected $\tilde{p}_a \approx 1$ behavior with oscillations at discontinuities.
  • Figure 4: Convergence factors $\tilde{p}_a$ for (a) WENO-Z and (b) MUSCL schemes.
  • Figure 5: Semi-analytical solutions of the Bjorken flow system \ref{['Bulk_Milne']}: (a) Energy density $\varepsilon_{SA}$ versus $\tau$ and (b) Bulk viscous pressure $\Pi_{SA}$ versus $\tau$. Initial conditions: $\tau_0 = 0.5$, $\varepsilon(\tau_0) = 100$, $\Pi(\tau_0) = 0$. Solutions obtained using second-order Runge-Kutta with $\Delta \tau = 1/400$.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Remark 3.1
  • Theorem 1
  • Remark 3.2
  • Remark 4.1
  • Theorem 2
  • proof
  • Remark 4.2
  • Theorem 3
  • proof
  • Remark 5.1
  • ...and 7 more