Global regularity versus finite-time singularity formation for quasi-semilinear second-order hyperbolic systems with exclusively linearly degenerate modes

For nonlinear second-order hyperbolic systems of the form $B^{00}(U)U_{tt} + C^j(U)U_{tx_j} + B^{jk}(U)U_{x_jx_k} = H(U, U_t, U_x)$ , whose principal coefficients depend only on the unknown $U$ and which are pointwise semi-strictly definitely hyperbolic, Freistühler (2024) proved that every Lax mode of the first-order formulation is linearly degenerate, and that this applies to recently studied causal models of dissipative relativistic fluid dynamics (Freistühler–Temple and Bemfica–Disconzi–Noronha theories). The note then asks whether solutions of these models for smooth initial data generally avoid finite-time singularity formation. The small-data regime near homogeneous states has since been settled positively (global existence and asymptotic stability), but for smooth data of arbitrary size the question remains open: it is unknown whether exclusive linear degeneracy, possibly together with the dissipative source $H$ , precludes finite-time loss of regularity, or whether a new singularity mechanism — such as the inter-family characteristic enveloping known from the linearly degenerate Chaplygin gas — can still produce breakdown. The problem asks for a rigorous resolution of this question for these relativistic fluid models.

Background

A quasilinear first-order hyperbolic system $V_t + A(V)V_x = 0$ evolves smooth initial data that frequently develop singularities (loss of differentiability, i.e. shock formation) in finite time. Following Lax (1957), a characteristic mode — a smooth pair of an eigenvalue (characteristic speed) $\lambda$ and eigenvector $R$ of the matrix $A$ — is called genuinely nonlinear at a state if the directional derivative of $\lambda$ in state space along $R$ is nonzero there, and linearly degenerate if that derivative vanishes identically (a Lax mode is such a characteristic pair, so named after Lax's 1957 classification). Genuine nonlinearity is the classical engine of singularity formation: Lax, John, and Liu showed that smooth solutions typically break down when a genuinely nonlinear mode is present, and Bärlin (2023) extended this to inhomogeneous first-order systems $V_t + A(V)V_x = G(V)$ with source terms, requiring only genuine nonlinearity and $G(V^*) = 0$ . The classical blowup mechanism, however, does not automatically transfer to second-order systems. Freistühler (2024) considers systems $B^{00}(U)U_{tt} + C^j(U)U_{tx_j} + B^{jk}(U)U_{x_jx_k} = H(U, U_t, U_x)$ — the principal (second-order) part is linear in the highest derivatives with coefficients depending only on $U$ , a structure he calls quasi-semilinear — under the assumption of pointwise semi-strict definite hyperbolicity (at each frozen state, $B^{00}$ is negative definite and the constant-coefficient linear operator has, for every unit spatial frequency on the unit sphere, only real nonzero roots $\xi_0$ of its dispersion polynomial — the polynomial in the temporal frequency $\xi_0$ obtained by freezing the coefficients — with multiplicities and amplitude-space dimensions independent of the spatial direction). Writing $P = U_t$ and $Q_j = U_{x_j}$ , the system is equivalent to a first-order system in $(P, Q_1, ..., Q_d, U)$ with a lower-order source. The key structural observation (Proposition 1 of the note) is that all modes of this first-order formulation are linearly degenerate: the nonzero characteristic speeds depend only on $U$ , while the associated eigenvectors have zero component in the $U$ -direction, so the derivative of the speed along the eigenvector vanishes; the zero-speed mode is trivially linearly degenerate. By Corollary 1 of the note, the first-order versions of the recently studied causal models of dissipative relativistic fluid dynamics — the Freistühler–Temple models of pure radiation, ideal gases, and barotropic fluids, and the Bemfica–Disconzi–Noronha (BDNK) class, in which the viscous energy-momentum tensor is built from first-order gradient terms so that the equations of motion form a causal second-order hyperbolic system — have exclusively linearly degenerate modes. The note closes with the question of whether solutions of these models for smooth data generally avoid singularity formation, analogously to what is expected for hyperbolic-parabolic composite systems such as the compressible Navier–Stokes–Fourier equations. The adjacent literature delimits both directions of this question. In one direction, linear degeneracy of all modes does not by itself preclude breakdown in general first-order systems: for the one-dimensional Chaplygin gas (the special isentropic gas whose sound speed times density is constant, making both characteristic fields linearly degenerate), Kong, Wei, and Zhang constructed finite-time singularities arising from the envelope of different characteristic families rather than from genuine nonlinearity within one family, and Bemfica (2025) proved finite-time gradient blow-up for Israel–Stewart-type dissipative relativistic theories in plane symmetry (Israel–Stewart theory is a causal second-order-in-gradients viscous formalism related to, but distinct from, the class considered here). In the other direction, the small-data theory is now positive: Sroczinski proved global-in-time existence and asymptotic stability of solutions to quasilinear second-order uniformly dissipative hyperbolic–hyperbolic systems (systems in which two hyperbolic operators interact dissipatively, without symmetry hypotheses, treated by para-differential calculus, a technique that analyzes variable-coefficient nonlinear operators as perturbations of frozen-coefficient linear ones), and together with Freistühler verified the required dissipativity for the relativistic models, giving nonlinear stability of homogeneous states; smooth shock profiles (traveling waves connecting end states) of arbitrary strength exist in the Freistühler–Temple barotropic model and are nonlinearly stable when spectrally stable. What lies between — smooth data of arbitrary size, where no smallness around a constant state is available and where the dissipative structure is a genuinely nonlinear perturbation — is the open territory of this problem.

Problem Statement

Consider systems of the form $B^{00}(U)U_{tt} + C^j(U)U_{tx_j} + B^{jk}(U)U_{x_jx_k} = H(U, U_t, U_x)$ (sums over repeated spatial indices $j, k$ understood) for $U: [0, T) \times \mathbb{R}^d \to \mathbb{R}^n$ , with the principal coefficients depending only on $U$ (not on derivatives of $U$ ) and pointwise semi-strictly definitely hyperbolic, in the following sense of the source note: at each frozen value of $U$ , the matrix $B^{00}$ is negative definite and the constant-coefficient linear operator has, for every unit spatial frequency, only real nonzero roots $\xi_0$ of its dispersion polynomial (the polynomial in the temporal frequency $\xi_0$ obtained by freezing the coefficients), with multiplicities and amplitude-space dimensions independent of the spatial direction. Writing $P = U_t$ and $Q_j = U_{x_j}$ , the system is equivalent to a first-order system in the variables $(P, Q_1, ..., Q_d, U)$ with a lower-order source, and by Freistühler's Proposition 1 every Lax mode — that is, every characteristic pair of eigenvalue and eigenvector of the first-order formulation's symbol — is linearly degenerate (the directional derivative of the eigenvalue along the eigenvector vanishes identically). This class includes the Freistühler–Temple and Bemfica–Disconzi–Noronha models of dissipative relativistic fluid dynamics, whose lower-order term $H$ is fixed by the physical model (and whose dissipative structure was verified by Freistühler–Sroczinski). Following the closing question of the source note, determine whether, for these relativistic fluid models, classical solutions arising from smooth initial data of arbitrary size (not restricted to small perturbations of homogeneous states, that regime being already settled positively by Sroczinski and by Freistühler–Sroczinski) exist globally in time: that is, does the exclusive linear degeneracy of the modes, possibly together with the dissipative source $H$ of these models, prevent finite-time loss of regularity, or does there exist smooth initial data for one of these models whose classical solution breaks down in finite time? A rigorous global-regularity proof covering all of these models, or exhibited breaking-down data for any one of them, settles the question; a characterization of the singularity mechanism in the event of breakdown (for example, enveloping between different characteristic families as in the linearly degenerate Chaplygin gas, multi-dimensional focusing, or source-driven amplification) and of the role of $H$ would be of additional interest but is not itself a required part of the resolution. The scientific objective is to settle global regularity versus finite-time singularity formation for smooth data of arbitrary size for these dissipative relativistic fluid models with exclusively linearly degenerate modes.

References

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