No shock formation for totally linearly degenerate quasilinear hyperbolic systems: the $C^0\Rightarrow C^1$ conjecture beyond block-structured classes

For one-dimensional strictly hyperbolic quasilinear systems in which every characteristic field is linearly degenerate in the sense of Lax, a long-standing conjecture (essentially stated in Majda's 1984 monograph and phrased by Brenier as "smooth, or blow up in Sup norm, not Lipschitz norm") asserts that no geometric, shock-type singularity can form: if the $C^1$ solution breaks down, the solution itself must escape bounded sets (an ODE-type singularity), but a uniform $C^0$ bound must force a uniform $C^1$ bound. Li, Peng, Yang and Zhou (2008) reduced this to an a priori implication (their Property A, $C^0\Rightarrow C^1$) and verified it for classes built from strict block hyperbolicity, part richness, and successively block-closed completely reducible architecture, in conservation form. The open problem is whether these structural hypotheses can be removed: prove or refute Property A, $C^0\Rightarrow C^1$, for totally linearly degenerate systems of conservation laws with arbitrary bounded $C^1$ initial data.

Background

Consider the Cauchy problem for a first-order quasilinear hyperbolic system in one space dimension, $\partial_t u + A(u)\,\partial_x u = 0, \qquad u|_{t=0}=\varphi(x),$ where $u\in\mathbb{R}^n$ and $A(u)$ is a smooth $n\times n$ matrix. Strict hyperbolicity means $A(u)$ has $n$ distinct real eigenvalues $\lambda_1(u)<\cdots<\lambda_n(u)$ (the characteristic speeds), with left and right eigenvectors $l_i(u)$, $r_i(u)$. Lax's classification calls the $i$-th characteristic field genuinely nonlinear if $\nabla\lambda_i(u)\cdot r_i(u)\neq 0$ (the speed varies along the flow direction of that family) and linearly degenerate (LD) if $\nabla\lambda_i(u)\cdot r_i(u)\equiv 0;$ the system is totally linearly degenerate if all fields are LD. Classical results of Lax (1964) show that genuine nonlinearity drives shock formation: characteristics of one family overtake each other, forming an envelope, along which the solution stays bounded but its first derivatives blow up — a geometric singularity. For LD fields the speeds are constant along their own characteristic flow, so this compression mechanism is absent: for diagonal systems, and for small and decaying initial data of general systems (the weakly-linearly-degenerate theory of Li and Kong, where $\nabla\lambda_i(u)\cdot r_i(u)\equiv 0$ is required only along the characteristic trajectory through a reference state), global classical solutions exist and LD systems never form singularities. The subtlety appears for arbitrary (large) $C^1$ data when the eigenvectors depend on $u$. Jeffrey's example is a strictly hyperbolic system with constant eigenvalues (hence LD) whose $C^1$ solution blows up in finite time with the solution $u$ itself becoming unbounded — an ODE-type singularity, driven by the state dependence of the eigenvectors rather than by characteristic compression (Li and Liu, 2003, systematically developed this eigenvector mechanism). Thus for LD systems one cannot hope to exclude blowup in general; Majda's 1984 monograph essentially conjectured, and Brenier (2004) phrased explicitly, that totally linearly degenerate systems of conservation laws are "smooth or blow up in Sup norm, not in Lipschitz norm": breakdown may occur only through escape of $u$ from bounded sets, never through a shock-type singularity with $u$ bounded and $\partial_x u$ blowing up along an envelope of same-family characteristics. Li, Peng, Yang and Zhou (2008) proposed a framework to justify this conjecture: it suffices to prove Property A, the implication $C^0\Rightarrow C^1$ — on any finite time interval, a uniform bound on the $C^0$ norm of the classical solution forces a uniform bound on its $C^1$ norm. They verified Property A, and hence absence of shock formation, for strictly block-hyperbolic systems (the eigenvalue list splits into blocks separated in speed) that are part rich (Lax's transformation relating shock and rarefaction curves remains valid within distinguished variable subspaces), successively block closed (an $m$-step completely reducible, triangular architecture in which lower-step subsystems are closed), and reducible to conservation-law form by a $C^2$ change of variables, under additional coefficient-constancy and interaction-coefficient pattern conditions. The same circle of ideas covers systems with source terms in special cases (Kerr–Debye models, relaxation systems).

Problem Statement

Let $\partial_t u + A(u)\,\partial_x u = 0$ be a one-dimensional strictly hyperbolic quasilinear system with smooth entries, all of whose characteristic fields are linearly degenerate ($\nabla\lambda_i(u)\cdot r_i(u)\equiv 0$ for $i=1,\dots,n$), taken in the class of systems reducible to conservation-law form $\partial_t u + \partial_x f(u)=0$ (the form in which Majda's conjecture is stated); let $\varphi\in C^1$ be arbitrary initial data with bounded $C^1$ norm, not assumed small or decaying. Prove or refute the long-standing conjecture (Majda 1984; Brenier 2004; formulated as the operative criterion by Li–Peng–Yang–Zhou 2008): the system satisfies Property A — for any given $T_0>0$ and every $T<T_0$, if the classical solution on $[0,T]$ admits a uniform a priori bound $\|u(t,\cdot)\|_{L^\infty}\le C_0(T_0)$, then it also admits a uniform a priori bound on the $C^1$ norm, $\|u_x(t,\cdot)\|_{L^\infty}\le C_1(T_0)$, where the constants are independent of $T$ but may depend on $T_0$, the fixed system, and the initial data — so that no geometric, shock-type singularity (bounded solution with first derivatives blowing up along an envelope of characteristics of a single family) can occur, and any finite-time breakdown must be of ODE type with the solution escaping bounded sets. A refutation consists of an explicit totally linearly degenerate system of conservation-law form and bounded-$C^1$ initial data whose classical solution stays uniformly $C^0$-bounded while $\|\partial_x u\|_{L^\infty}$ blows up in finite time. This is the open problem in the general situation left by Li–Peng–Yang–Zhou: whether the $C^0\Rightarrow C^1$ mechanism extends beyond the classes for which it is known — to systems that are not block reducible (non-completely reducible) — without the structural hypotheses of strict block hyperbolicity, part richness, successive block closure (complete reducibility), and the triangular interaction-coefficient pattern.

References

1. Ta-Tsien Li, Yue-Jun Peng, Yong-Fu Yang, Yi Zhou, "Mechanism of the formation of singularities for quasilinear hyperbolic systems with linearly degenerate characteristic fields", Mathematical Methods in the Applied Sciences 31 (2008), no. 2, 193–227. DOI 10.1002/mma.902, https://doi.org/10.1002/mma.902
2. Tatsien Li (Daqian Li), "Mechanism of the formation of singularities for quasilinear hyperbolic systems", International Journal of Numerical Analysis and Modeling 6 (2009), no. 3, 520–531, http://www.math.ualberta.ca/ijnam/Volume-6-2009/No-3-09/2009-03-11.pdf (zbMATH 5543995)
3. Andrew Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables", Applied Mathematical Sciences 53, Springer, New York, 1984. DOI 10.1007/978-1-4612-1116-7, https://doi.org/10.1007/978-1-4612-1116-7 (ISBN 978-1-4612-1116-7)
4. Yann Brenier, "Hydrodynamic structure of the augmented Born-Infeld equations", Archive for Rational Mechanics and Analysis 172 (2004), no. 1, 65–91. DOI 10.1007/s00205-003-0291-4, https://doi.org/10.1007/s00205-003-0291-4
5. Peter D. Lax, "Development of singularities of solutions of nonlinear hyperbolic partial differential equations", Journal of Mathematics and Physics 5 (1964), 611–613. DOI 10.1063/1.1704154, https://doi.org/10.1063/1.1704154
6. A. Jeffrey, "Quasilinear Hyperbolic Systems and Waves", Research Notes in Mathematics 5, Pitman Publishing, London–San Francisco–Melbourne, 1976. zbMATH Zbl 0322.35060, https://zbmath.org/?q=an%3A3503761
7. Ta-tsien Li, De-xing Kong, "Breakdown of classical solutions to quasilinear hyperbolic systems", Nonlinear Analysis, Theory, Methods & Applications, Series A 40 (2000), 407–437. DOI 10.1016/S0362-546X(00)85025-7, https://doi.org/10.1016/S0362-546X(00)85025-7
8. Ta-Tsien Li, Fa-Gui Liu, "Singularity Caused by Eigenvectors for Quasilinear Hyperbolic Systems", Communications in Partial Differential Equations 28 (2003), no. 3–4, 477–503. DOI 10.1081/PDE-120020484, https://doi.org/10.1081/PDE-120020484
9. Peng Qu, "Mechanism of singularity formation for quasilinear hyperbolic systems with linearly degenerate characteristic fields", Journal of Differential Equations 251 (2011), no. 8, 2066–2081. DOI 10.1016/j.jde.2011.07.005, https://doi.org/10.1016/j.jde.2011.07.005
10. Peng Qu, "ODE singularity for blocked linearly degenerate quasilinear hyperbolic systems", Mathematical Methods in the Applied Sciences 34 (2011), no. 8, 950–957. DOI 10.1002/mma.1413, https://doi.org/10.1002/mma.1413
11. Peng Qu, "Blow-up of periodic solutions to linearly degenerate quasilinear hyperbolic systems", Nonlinear Analysis, Theory, Methods & Applications, Series A 75 (2012), 6520–6523. DOI 10.1016/j.na.2012.07.027, https://doi.org/10.1016/j.na.2012.07.027
12. G. Carbou, B. Hanouzet, R. Natalini, "Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation", Journal of Differential Equations 246 (2009), no. 1, 291–319. DOI 10.1016/j.jde.2008.05.015, https://doi.org/10.1016/j.jde.2008.05.015
13. Ta-Tsien Li, Yue-Jun Peng, Jérémy Ruiz, "Entropy solutions for linearly degenerate hyperbolic systems of rich type", Journal de Mathématiques Pures et Appliquées 91 (2009), no. 6, 553–568. DOI 10.1016/j.matpur.2009.01.008, https://doi.org/10.1016/j.matpur.2009.01.008 (rich system: shock and rarefaction curves coincide)
14. Yue-Jun Peng, Yong-Fu Yang, "Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type", Discrete and Continuous Dynamical Systems 35 (2015), no. 8, 3683–3706. DOI 10.3934/dcds.2015.35.3683, https://doi.org/10.3934/dcds.2015.35.3683
15. Yong-Fu Yang, "Mechanism of the formation of singularities for diagonal systems with linearly degenerate characteristic fields", Communications on Pure and Applied Analysis 8 (2009), no. 2, 757–768. DOI 10.3934/cpaa.2009.8.757, https://doi.org/10.3934/cpaa.2009.8.757
16. De-Xing Kong, Changhua Wei, Qiang Zhang, "Formation of singularities in one-dimensional Chaplygin gas", preprint (2013). arXiv:1311.3474, https://arxiv.org/abs/1311.3474
17. Heinrich Freistühler, "Linear Degeneracy in a Class of Nonlinear Second-Order Hyperbolic Systems", preprint (2024). arXiv:2406.00719, https://arxiv.org/abs/2406.00719
18. Changhua Wei, Tiantian Xing, "Global Classical Solution and Asymptotic Behavior to a Kind of Linearly Degenerate Quasilinear Hyperbolic System", Annals of Applied Mathematics 40 (2024), no. 3, 314–332. DOI 10.4208/aam.OA-2024-0017, https://doi.org/10.4208/aam.OA-2024-0017