Global regularity versus finite-time singularity formation in linearly degenerate dissipative relativistic fluid models

Freistühler (2024) proved that all Lax modes of the natural first-order reduction of quasisemilinear second-order hyperbolic systems of the form $B^{00}(U)U_{tt}+C^{j}(U)U_{tx_{j}}+B^{jk}(U)U_{x_{j}x_{k}}=H(U,U_{t},U_{x})$ are linearly degenerate, observed that this applies to the then-recent models of dissipative relativistic fluid dynamics of Freistühler–Temple and Bemfica–Disconzi–Noronha type, and closed his note with the question of whether solutions of these models for smooth initial data generally avoid finite-time singularity formation. The question remains open. Blowup is now known for other members of the same quasisemilinear class (the one-dimensional family $u_{tt}=c(u)^{2}u_{xx}+\lambda c(u)c'(u)(u_{x})^{2}$ for $0<\lambda\le2$, including the variational wave equation), so exclusive linear degeneracy by itself does not prevent singularity formation; and dissipative relativistic theories with genuinely nonlinear modes (Israel–Stewart theory) provably form shocks from smooth data. For the linearly degenerate dissipative relativistic models themselves, only small-data results (global existence and nonlinear stability near homogeneous states) and favorable numerical experiments in the hydrodynamic-frame-robust regime are available. The problem is to settle the large-data regularity status of these models: prove global-in-time regularity for general smooth data or construct smooth-data finite-time breakdown, with a characterization of the mechanism — in particular the role of the dissipative source $H$ in promoting or preventing singularity formation — as a complementary objective.

Background

A first-order quasilinear hyperbolic system $V_{t}+A(V)V_{x}=0$ in one space dimension has, at each state $V$, characteristic speeds and directions given by the eigenpairs $(\lambda,R)$ of the flux matrix $A$. Following Lax, a mode (an eigenvalue–eigenvector pair) is called genuinely nonlinear when the derivative of the speed $\lambda$ in the direction of its eigenvector $R$ is nonzero, and linearly degenerate when this directional derivative vanishes identically. Genuine nonlinearity is the engine of the classical shock-formation theorems of Lax, John, and Liu: smooth solutions generically develop unbounded first derivatives in finite time. Bärlin (2023) extended these blowup results to inhomogeneous systems $V_{t}+A(V)V_{x}=G(V)$ with quite general source terms $G$. Systems whose modes are all linearly degenerate are not covered by any general blowup theorem — but neither by any general global-regularity theorem.

The present question concerns second-order systems. Freistühler–Temple (2014 onward) and Bemfica–Disconzi–Noronha (2018 onward) introduced causal first-order formulations of dissipative (viscous, heat-conductive) relativistic fluid dynamics — the framework often called BDNK in the physics literature. In these models the dissipative corrections to the stress-energy tensor depend on gradients of the primary variables, so the field equations, written in terms of those variables, become second-order quasisemilinear hyperbolic systems

$$B^{00}(U)U_{tt}+C^{j}(U)U_{tx_{j}}+B^{jk}(U)U_{x_{j}x_{k}}=H(U,U_{t},U_{x}),$$

where $U$ collects the unknowns, the coefficient matrices depend only on $U$ and not on its derivatives (this is what 'quasisemilinear' means here), and $H$ is a lower-order source containing the dissipative terms. The systems are assumed (semi-strictly) definitely hyperbolic: at every state, the dispersion relation $\det(\xi_{0}^{2}B^{00}+\xi_{0}C(\xi)+B(\xi))=0$ has only real, nonzero zeros whose multiplicities do not depend on the spatial wave vector $\xi$ and equal the dimensions of the corresponding amplitude spaces. Physically, the expectation is an analogy with the hyperbolic-parabolic Navier–Stokes–Fourier system of compressible fluid dynamics, whose diffusion prevents shock formation, rather than with inviscid relativistic Euler flow, which forms shocks.

Introducing $P=U_{t}$, $Q_{j}=U_{x_{j}}$, and $U$ itself as variables yields the first-order formulation $A^{0}(V)V_{t}+A^{k}(V)V_{x_{k}}=H(V)$ with state $V=(P,Q_{1},\ldots,Q_{d},U)$. Freistühler (2024) proved that every mode of this first-order formulation is linearly degenerate: the nonzero characteristic speeds are the zeros of the second-order dispersion relation and depend only on $U$, while the associated eigenvectors have zero component in the $U$-direction, so the derivative of each speed along its own amplitude space vanishes. In particular, all the first-order versions of the dissipative relativistic fluid models listed above have exclusively linearly degenerate modes — the classical Lax mechanism of shock formation is absent from them — and Freistühler's note ends by asking whether their solutions for smooth data therefore generally avoid singularity formation.

Linear degeneracy alone, however, does not settle regularity. The one-parameter family $u_{tt}=c(u)^{2}u_{xx}+\lambda c(u)c'(u)(u_{x})^{2}$ with $0\le\lambda\le2$ (here $\lambda$ denotes the family parameter, not a characteristic speed) lies in the same quasisemilinear class and likewise has exclusively linearly degenerate modes in its natural $(P,Q,U)$ first-order reduction, yet finite-time blowup of classical solutions is known throughout $0<\lambda\le2$: at $\lambda=1$ this is the variational wave equation (the Euler–Lagrange equation of a family of liquid-crystal energies, whose weak solutions are known to develop cusps and wave breaking), at $\lambda=2$ it is the p-system (the pair $v_{t}-u_{x}=0$, $u_{t}+p(v)_{x}=0$ of Lagrangian compressible flow) in second-order form, whose blowup is the classical shock formation of the genuinely nonlinear $2\times2$ p-system — a first-order form different from the natural reduction — and Sugiyama (2026) proved blowup for the intermediate values $1<\lambda<2$ as well; Hu and Wang (2019) proved finite-time breakdown of smooth solutions for the multidimensional variational wave equation even for arbitrarily small initial energy. What distinguishes the dissipative relativistic models is the combination of the linearly degenerate mode structure with a dissipative source $H$. Whether that combination suffices to prevent singularity formation for smooth data of arbitrary size is exactly what is not known. Nearby evidence cuts both ways: Israel–Stewart theory, an alternative dissipative relativistic framework in which the dissipative fluxes obey relaxation equations and whose first-order modes are genuinely nonlinear, has recently been shown by Bemfica (2025) to form shocks from smooth data in 1+1 dimensions; while for the linearly degenerate models themselves, Sroczinski and Freistühler–Sroczinski proved global existence and nonlinear stability for small perturbations of homogeneous states, and the numerical experiments of Clarisse et al. (2025) found no shock formation in the BDNK model within its hydrodynamic-frame-robust regime — the regime of sufficiently small Knudsen number (the ratio of mean free path to macroscopic length scale) in which the theory's predictions do not depend on the choice of hydrodynamic frame, that is, on how the dissipative terms are split between the stress-energy tensor and its sources.

Problem Statement

For the dissipative relativistic fluid-dynamics models of Freistühler–Temple type and Bemfica–Disconzi–Noronha type — each of which is a quasisemilinear second-order hyperbolic system

$$B^{00}(U)U_{tt}+C^{j}(U)U_{tx_{j}}+B^{jk}(U)U_{x_{j}x_{k}}=H(U,U_{t},U_{x})$$

whose coefficient matrices depend only on $U$, whose first-order reductions have exclusively linearly degenerate modes, and whose source terms $H$ encode viscous and heat-conductive dissipation — determine whether smooth initial data can lead to finite-time singularity formation, as asked at the close of Freistühler's note: could it be that solutions of these models for smooth data generally avoid singularity formation? Concretely: (i) prove, for one or more of these models, global-in-time existence of classical solutions for general smooth initial data of arbitrary size, or construct, for at least one such model, smooth initial data whose classical solution ceases to be differentiable in finite time; and (ii) in either outcome, a characterization of the mechanism — in particular of whether the dissipative source $H(U,U_{t},U_{x})$ prevents or promotes breakdown, and of which mechanisms other than Lax genuine nonlinearity (such as wave-interaction steepening or source-driven amplification of derivatives, the mechanisms that produce blowup elsewhere in the quasisemilinear class) can drive or preclude breakdown — would complement a resolution of (i) rather than being required to answer the regularity question posed there. Answers restricted to small perturbations of homogeneous equilibrium states do not settle the question as posed, since small-data stability of these models is already established.

References

1. Heinrich Freistühler, Linear Degeneracy in a Class of Nonlinear Second-Order Hyperbolic Systems, arXiv:2406.00719 (2024), DOI 10.48550/arXiv.2406.00719, https://arxiv.org/abs/2406.00719 (source of the problem)
2. Heinrich Freistühler and Matthias Sroczinski, Nonlinear Stability of First-Order Relativistic Viscous Hydrodynamics, arXiv:2406.00673 (2024), DOI 10.48550/arXiv.2406.00673, https://arxiv.org/abs/2406.00673
3. Fabio S. Bemfica, Marcelo M. Disconzi, and Jorge Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Physical Review D 98, 104064 (2018), DOI 10.1103/PhysRevD.98.104064, arXiv:1708.06255, https://arxiv.org/abs/1708.06255
4. Heinrich Freistühler and Blake Temple, Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation, Proceedings of the Royal Society A 470, 20140055 (2014), DOI 10.1098/rspa.2014.0055, https://doi.org/10.1098/rspa.2014.0055
5. Johannes Bärlin, Formation of singularities in solutions to nonlinear hyperbolic systems with general sources, Nonlinear Analysis: Real World Applications 73, 103901 (2023), DOI 10.1016/j.nonrwa.2023.103901, arXiv:2204.12101, https://doi.org/10.1016/j.nonrwa.2023.103901
6. Matthias Sroczinski, Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems, arXiv:2301.01685 (2023), DOI 10.48550/arXiv.2301.01685, https://arxiv.org/abs/2301.01685
7. Matthias Sroczinski, Global existence and asymptotic decay for small solutions of general quasilinear hyperbolic balance laws, arXiv:2510.09318 (2025), DOI 10.48550/arXiv.2510.09318, https://arxiv.org/abs/2510.09318
8. Nicolas Clarisse, Eduardo O. Pinho, Teerthal Patel, Fabio S. Bemfica, Mauricio Hippert, and Jorge Noronha, Flux-Conservative BDNK Hydrodynamics and Shock Regularization, arXiv:2510.16603 (2025), DOI 10.48550/arXiv.2510.16603, https://arxiv.org/abs/2510.16603
9. Fábio S. Bemfica, Finite-time gradient blow-up and shock formation in Israel-Stewart theory: Bulk, shear, and diffusion regimes, Physical Review E 112, 065105 (2025), DOI 10.1103/blhw-xplr, arXiv:2508.04717, https://arxiv.org/abs/2508.04717
10. Yuusuke Sugiyama, Formation of singularities for a family of one-dimensional quasilinear wave equations beyond the variational case, arXiv:2606.23605 (2026), DOI 10.48550/arXiv.2606.23605, https://arxiv.org/abs/2606.23605
11. Sondre Tesdal Galtung and Katrin Grunert, Wave breaking for the nonlinear variational wave equation, arXiv:2502.19889 (2025), DOI 10.48550/arXiv.2502.19889, https://arxiv.org/abs/2502.19889
12. Yanbo Hu and Guodong Wang, Singularity for a multidimensional variational wave equation arising from nematic liquid crystals, arXiv:1910.08671 (2019), DOI 10.48550/arXiv.1910.08671, https://arxiv.org/abs/1910.08671
13. Heinrich Freistühler, Formation of Singularities in Solutions to Ruggeri's Hyperbolic Navier-Stokes Equations, arXiv:2305.05426 (2023), DOI 10.48550/arXiv.2305.05426, https://arxiv.org/abs/2305.05426