For one-dimensional first-order quasilinear hyperbolic systems whose characteristic fields are all linearly degenerate, a long-standing conjecture — implicit in the 1984 Springer monograph on compressible flow and conservation laws, discussed by Brenier (2004), and formalized as Property A by Li, Peng, Yang and Zhou (2008) — asserts that no geometric shock formation can occur: as long as a classical (C1C^1) solution stays uniformly bounded, its first derivatives stay uniformly bounded, so any finite-time breakdown must proceed from a non-geometric (e.g., ODE-type) mechanism rather than characteristic intersection. The conjecture is proved only for restricted structural classes (diagonalizable, constant-multiplicity, block-structured 'rich' systems). The open problem is to prove or refute Property A for general such systems — including systems that are not block-reducible, with non-constant eigenvalues and with general source terms.

Background

A first-order quasilinear hyperbolic system in one space dimension is a Cauchy problem of the form ∂tu+∂xf(u)=g(u)\partial_t u+\partial_x f(u)=g(u), or more generally ∂tu+A(u)∂xu=F(u)\partial_t u+A(u)\partial_x u=F(u), where u=(u1,…,un)u=(u_1,\dots,u_n) is the unknown state vector and the n×nn\times n matrix A(u)A(u) has real eigenvalues λ1(u)≤⋯≤λn(u)\lambda_1(u)\le\dots\le\lambda_n(u) together with a corresponding basis of right eigenvectors (hyperbolicity); it is strictly hyperbolic when the eigenvalues are distinct. Each eigenvalue–eigenvector pair (λi,ri)(\lambda_i,r_i) is called a characteristic field: through each point passes an ii-characteristic curve dx/dt=λi(u)dx/dt=\lambda_i(u). The standard classification of characteristic fields distinguishes genuinely nonlinear fields, for which the eigenvalue changes along the field direction (∇λi⋅ri≠0\nabla\lambda_i\cdot r_i\neq0), from linearly degenerate fields, for which ∇λi⋅ri≡0\nabla\lambda_i\cdot r_i\equiv0 identically in state space, so the eigenvalue is constant along the corresponding contact-discontinuity curves. For genuinely nonlinear fields — the prototype being Burgers' equation — classical solutions of typical Cauchy data break down in finite time by geometric shock formation: characteristics of one family intersect, the waveform steepens, and the spatial derivative blows up while the solution itself remains bounded. The classical picture suggests that linear degeneracy should remove exactly this mechanism: for scalar equations and diagonal systems, linearly degenerate characteristics are straight lines along which no self-focusing occurs, and small-data global regularity holds. Whether the same is true for general coupled systems is a long-standing open question, discussed in Majda's 1984 Springer monograph on compressible flow and conservation laws (Applied Mathematical Sciences vol. 53), by Brenier in his study of the augmented Born–Infeld equations (a linearly degenerate model related to the Chaplygin gas, i.e., a fluid with equation of state of the form p=P0−D/ρp=P_0-D/\rho), and stated as an explicit conjecture by Li, Peng, Yang and Zhou (2008): for the Cauchy problem with all characteristic fields linearly degenerate, no geometric shock formation occurs. Li–Peng–Yang–Zhou formalized this as Property A: along the life span of a C1C^1 solution, a uniform bound on the solution (a C0C^0 bound) implies a uniform bound on its spatial derivatives (a C1C^1 bound), so characteristic intersection can never be forced by wave interaction while the amplitude stays bounded. They proved Property A for classes of systems satisfying a hierarchy of structural hypotheses — strict block hyperbolicity (the eigenvalues cluster into separated blocks inside which eigenvalues may coincide), part richness (an integrability condition ensuring that each block admits enough Riemann invariants — scalar functions of the state that stay constant along all but one characteristic family, yielding adapted coordinates in state space), a successively block-closed (completely reducible) architecture, a triangular/vanishing pattern for the interaction coefficients between families, and coefficient-constancy conditions — and they explicitly left the conjecture open without these hypotheses. Partial results outside their framework cover only special cases: 2×2 systems (Kong–Tsuji 1999), systems with characteristics of constant multiplicity and small total-variation data (Dai–Kong 2007), and diagonal or rich systems. Meanwhile later work showed that linear degeneracy does not prevent all breakdown: Qu (2011, 2012) proved that C1C^1 solutions of linearly degenerate systems can still blow up by an ODE-type mechanism (all W1,pW^{1,p} norms blowing up simultaneously, driven by the source/amplitude dynamics rather than wave geometry), and Kong, Wei and Zhang (2014) showed that the one-dimensional Chaplygin gas — a linearly degenerate system — develops delta-shock-type concentration singularities for large data through envelopes of different characteristic families, with the solution amplitude becoming unbounded. These results delimit what the conjecture can mean: singularities may occur, but, conjecturally, never through the geometric, same-family characteristic-intersection mechanism, provided the solution itself remains bounded. Whether this is true for general linearly degenerate systems — not block-reducible, with non-constant eigenvalues, and with general source terms — remains unresolved, and the largest structural class for which the conjecture holds is not known.

Problem Statement

Consider the Cauchy problem for first-order quasilinear hyperbolic systems ∂tu+A(u)∂xu=F(u)\partial_t u+A(u)\partial_x u=F(u) in one space dimension, with A(u)A(u) having real eigenvalues, such that every characteristic field is linearly degenerate in the sense that each field admits locally smooth eigenvalue branches and eigenvector fields satisfying ∇λi(u)⋅ri(u)≡0\nabla\lambda_i(u)\cdot r_i(u)\equiv0 for all ii — in particular strictly hyperbolic systems (λ1(u)<⋯<λn(u)\lambda_1(u)<\dots<\lambda_n(u)), optionally admitting constant-multiplicity eigenvalues with linear degeneracy imposed within each block as in Dai–Kong (2007) and Peng–Yang (2012). The conjecture of Li–Peng–Yang–Zhou, tracing back to Majda's 1984 conservation-laws monograph and Brenier's 2004 work, is that such systems admit no geometric shock formation: if the C1C^1 solution exists on a time interval [0,T)[0,T) and remains uniformly bounded there, then its spatial derivatives remain uniformly bounded on [0,T)[0,T) (Property A), so no characteristic intersection of the shock type can occur while the amplitude stays bounded. Determine whether Property A holds in this generality — including systems that are not block-reducible (not completely reducible), systems with non-constant eigenvalues, and systems with general source terms F(u)F(u) — without assuming the structural hypotheses under which the conjecture has been proved (strict block hyperbolicity, part richness, successively block-closed (completely reducible) architecture, triangular/vanishing interaction coefficients, and coefficient-constancy conditions).

References

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