Majda's ODE-type singularity conjecture for linearly degenerate quasilinear hyperbolic systems

For one-dimensional quasilinear hyperbolic systems whose characteristic fields are all linearly degenerate, a long-standing conjecture attributed to A. Majda asserts that every finite-time breakdown of a classical $C^1$ solution must be an ODE-type singularity: the $L^\infty$ norm of the state itself blows up at the first singular time, rather than only its space-time derivatives as in shock formation for genuinely nonlinear fields. The conjecture has been verified only for special block-structured system classes and under conditional a priori bounds (Li–Peng–Yang–Zhou 2008; Qu 2011), while all known linearly degenerate systems that do break down in finite time — periodic counterexamples, the Chaplygin gas, a shallow-water magnetohydrodynamics system — blow up in amplitude, so the general blow-up mechanism remains undetermined. The problem is to prove or disprove the conjecture for general linearly degenerate quasilinear hyperbolic systems without block-structure assumptions and without a priori $L^\infty$/derivative bounds.

Background

A first-order quasilinear hyperbolic system in one space variable is a system $\partial_t u + A(u)\,\partial_x u = 0,\qquad u(x,0)=u_0(x),$ where $u$ takes values in $\mathbb{R}^n$ and $A(u)$ is a smooth real $n\times n$ matrix that is hyperbolic (all eigenvalues $\lambda_1(u),\dots,\lambda_n(u)$ real; strictly hyperbolic if they are distinct), with left and right eigenvectors $l_i(u)$, $r_i(u)$. The $i$-th characteristic field is called genuinely nonlinear (Lax) if $\nabla_u\lambda_i\cdot r_i\not\equiv 0$, and linearly degenerate if $\nabla_u\lambda_i\cdot r_i\equiv 0$: along each $i$-characteristic curve $\dot{x}=\lambda_i(u)$, the characteristic speed is then constant to first order, so the field transports rather than compresses. For genuinely nonlinear systems (e.g. the $p$-system of isentropic gas dynamics), classical solutions of typical smooth Cauchy data form shocks in finite time by the Lax–John theory: the solution $u$ stays bounded while $\partial_x u$ blows up — a geometric, shock-type singularity caused by the intersection of characteristics of the same family. For linearly degenerate fields (contact discontinuities in gas dynamics, the Chaplygin gas with equation of state $p=-A/\rho$, Born–Infeld-type electrodynamics, chromatography systems) no such self-compression occurs, and A. Majda's 1984 monograph on compressible flow and conservation laws conjectured that linearly degenerate systems cannot form shocks: if a classical solution breaks down at all, the singularity must be of ODE type, i.e. the amplitude $\|u(\cdot,t)\|_{L^\infty}$ itself blows up as $t$ approaches the first singular time $T^*$ (the supremum of existence times of the $C^1$ solution), the state being driven to infinity by characteristic-wise ODE dynamics rather than by geometric compression. The question goes back to Jeffrey's 1974 study of completely exceptional (linearly degenerate) systems, and was re-posed by Brenier in 2004 for the augmented Born–Infeld equations. A program initiated by Li, Peng, Yang and Zhou (2008) introduced block-structural frameworks — strict block hyperbolicity, part richness, successively block-closed (completely reducible) systems — in which the ODE-type conclusion was proved, and Peng Qu (2011) removed the part-richness hypothesis for blocked systems, extended the result to constant-multiplicity second blocks, and gave an energy-method mechanism in the $C^1$ framework reducing the general conjecture to a purely analytic statement (Conjecture 2: a uniform $L^p$ bound, $1<p\le\infty$, on $\partial_x u$ under a uniform $L^\infty$ bound on $u$). However, all of these positive results are conditional — they assume a priori uniform $L^\infty$ bounds on $u$ and control of derivatives of part of the unknown — and heavily structural. Meanwhile finite-time breakdown does occur for linearly degenerate systems: Qu (2012) constructed strictly hyperbolic linearly degenerate systems with $n>2$ and small periodic data whose classical solutions blow up in finite time, and for the linearly degenerate Chaplygin gas, Kong–Wei–Zhang (2014) described cusp and delta-type singularities (with the density tending to infinity) produced by the envelope of different characteristic families. In every known example the amplitude blows up, which is consistent with the conjecture but does not prove it; whether a linearly degenerate system can exhibit shock-type (gradient-only) blow-up with $u$ uniformly bounded remains unknown. Qu's energy-method mechanism in the $C^1$ framework shows that all $W^{1,p}$ norms ($1\le p\le\infty$) of a $C^1$ solution breaking down at finite $T^*$ blow up simultaneously; consequently a uniform $L^p$ bound ($1<p\le\infty$) on $\partial_x u$ under a uniform $L^\infty$ bound on $u$ for general linearly degenerate systems without block-structure assumptions would be a sufficient condition for the conjecture (Qu's Conjecture 2), while such an estimate established only for a proper subclass of linearly degenerate systems would not by itself settle the conjecture as posed.

Problem Statement

Consider the Cauchy problem in one space variable $\partial_t u + A(u)\,\partial_x u = 0,\qquad u(\cdot,0)=u_0\in C^1,$ with $A$ smooth and hyperbolic (the eigenvalue/eigenvector fields $\lambda_i$, $r_i$ smooth on the state region under consideration, so that the class covers both strictly hyperbolic systems and systems of constant eigenvalue multiplicity), and assume that every characteristic field is linearly degenerate: $\nabla_u\lambda_i(u)\cdot r_i(u)\equiv 0$ for all $i$. Let $T^*>0$ be the maximal existence time of the classical $C^1$ solution, suppose $T^*<\infty$, and let $u$ remain a classical solution on $[0,T^*)$.

Majda's ODE-type singularity conjecture asserts that the breakdown must be of ODE type: $\lim_{t\uparrow T^*}\,\|u(\cdot,t)\|_{L^\infty}=+\infty.$ Determine whether this is true in general. Specifically:

1. Prove the conjecture for general linearly degenerate quasilinear hyperbolic systems — without block-structure assumptions (blocked/completely reducible/successively block-closed form, strict block hyperbolicity, constant-multiplicity second blocks) and without a priori hypotheses such as a uniform $L^\infty$ bound on $u$ combined with control of $\partial_x u^{(1)}$.

2. Or refute it: construct an explicit linearly degenerate system (strictly hyperbolic or of constant eigenvalue multiplicity) and $C^1$ initial data such that the classical solution blows up at some finite $T^*$ while $u$ stays uniformly bounded on $[0,T^*)$ and some derivative norm ($\partial_x u$ or $\partial_t u$) diverges — a shock-type/geometric singularity in a linearly degenerate system.

References

1. Alan Jeffrey, "Breakdown of the solution to a completely exceptional system of hyperbolic equations", Journal of Mathematical Analysis and Applications 45, No. 2 (1974), 375–381. DOI: 10.1016/0022-247X(74)90079-1, https://doi.org/10.1016/0022-247X(74)90079-1
2. Andrew J. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables", Applied Mathematical Sciences 53, Springer-Verlag, New York, 1984. DOI: 10.1007/978-1-4612-1116-7, https://doi.org/10.1007/978-1-4612-1116-7
3. Yann Brenier, "Hydrodynamic structure of the augmented Born-Infeld equations", Archive for Rational Mechanics and Analysis 172, No. 1 (2004), 65–91. DOI: 10.1007/s00205-003-0291-4, https://doi.org/10.1007/s00205-003-0291-4
4. 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, No. 2 (2008), 193–227. DOI: 10.1002/mma.902, https://doi.org/10.1002/mma.902
5. Peng Qu, "Mechanism of singularity formation for quasilinear hyperbolic systems with linearly degenerate characteristic fields", Journal of Differential Equations 251, No. 8 (2011), 2066–2081. DOI: 10.1016/j.jde.2011.07.005, https://doi.org/10.1016/j.jde.2011.07.005
6. Peng Qu, "ODE singularity for blocked linearly degenerate quasilinear hyperbolic systems", Mathematical Methods in the Applied Sciences 34, No. 8 (2011), 950–957. DOI: 10.1002/mma.1413, https://doi.org/10.1002/mma.1413
7. Peng Qu, "Blow-up of periodic solutions to linearly degenerate quasilinear hyperbolic systems", Nonlinear Analysis: Theory, Methods & Applications 75, No. 18 (2012), 6520–6523. DOI: 10.1016/j.na.2012.07.027, https://doi.org/10.1016/j.na.2012.07.027
8. De-Xing Kong, Changhua Wei, Qiang Zhang, "Formation of singularities in one-dimensional Chaplygin gas", Journal of Hyperbolic Differential Equations 11, No. 3 (2014), 521–561. DOI: 10.1142/S0219891614500155, https://doi.org/10.1142/S0219891614500155 (preprint arXiv:1311.3474, https://arxiv.org/abs/1311.3474)
9. Geng Chen, Tao Huang, Chun Liu, "Finite Time Singularities for Hyperbolic Systems", SIAM Journal on Mathematical Analysis 47, No. 1 (2015), 758–785. DOI: 10.1137/140986359, https://doi.org/10.1137/140986359
10. Dongyi Wei, Zhifei Zhang, Wenbin Zhao, "Global existence of the two-dimensional axisymmetric Euler equations for the Chaplygin gas with large angular velocities", Advanced Nonlinear Studies 22, No. 1 (2022), 635–658. DOI: 10.1515/ans-2022-0031, https://doi.org/10.1515/ans-2022-0031
11. Bingbing Ding, Zhouping Xin, Huicheng Yin, "On global smooth solutions to the 2D isentropic and irrotational Chaplygin gases with short pulse data", Advances in Mathematics 476 (2025), 110362. DOI: 10.1016/j.aim.2025.110362, https://doi.org/10.1016/j.aim.2025.110362 (preprint arXiv:2204.05017, https://arxiv.org/abs/2204.05017)
12. Yanbo Hu, Ying Zeng, "Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics", Advances in Nonlinear Analysis 14 (2025), Article 20250126. DOI: 10.1515/anona-2025-0126, https://doi.org/10.1515/anona-2025-0126
13. Heinrich Freistuhler, "Linear Degeneracy in a Class of Nonlinear Second-Order Hyperbolic Systems", preprint (2024). arXiv:2406.00719, https://arxiv.org/abs/2406.00719