Majda's conjecture: ODE-type singularity formation for one-dimensional quasilinear hyperbolic systems with linearly degenerate characteristic fields

For one-dimensional strictly hyperbolic quasilinear systems $\partial_t u+A(u)\,\partial_x u=F(u)$ whose characteristic fields are all linearly degenerate in Lax's sense, it is conjectured (essentially due to Majda, 1984) that any finite-time breakdown of the classical $C^1$ solution must be of ODE type: the amplitude $\|u(t,\cdot)\|_{L^\infty}$ itself must blow up at the lifespan, and a shock-type, gradient-only breakdown can never occur. Equivalently, a uniform $L^\infty$ bound on $u$ should imply a uniform $W^{1,p}$ bound on $u$ (i.e. a uniform $L^p$ bound on $\partial_x u$, $1<p\le\infty$) with constants depending only on the system and the time horizon. The conjecture is verified only for structured subclasses (diagonal, completely reducible/block systems, relaxation models such as Kerr–Debye); the general case — proof or counterexample — remains open.

Background

A first-order quasilinear hyperbolic system in one space dimension is a system $\partial_t u+A(u)\,\partial_x u=F(u)$ for an unknown vector $u(t,x)\in\mathbb{R}^n$, where $A(u)$ is an $n\times n$ matrix depending smoothly on $u$ and $F(u)$ is a source term (the case $F\equiv 0$ covers systems of conservation laws, Majda's original setting). The system is strictly hyperbolic if $A(u)$ has $n$ distinct real eigenvalues $\lambda_1(u)<\cdots<\lambda_n(u)$; the integral curves of $dx/dt=\lambda_i(u)$ are the characteristics of the $i$-th family, called the $i$-th characteristic field. Lax's genuine nonlinearity condition is $\nabla\lambda_i(u)\cdot r_i(u)\neq 0$ (with $r_i$ the right eigenvector): characteristic speed varies along the flow, characteristics of one family compress, and this is the classical shock-formation mechanism — for genuinely nonlinear systems (Lax, John) the $C^1$ solution typically breaks down in finite time through $\|\partial_x u\|_\infty\to\infty$ while $\|u\|_\infty$ stays bounded, as in Burgers' equation. The opposite condition, $\nabla\lambda_i(u)\cdot r_i(u)\equiv 0$ for every $i$, is linear degeneracy (LD); a system all of whose fields are LD is totally linearly degenerate. For such systems characteristics of a given family do not compress, so the classical shock mechanism is absent, and for diagonal or rich systems (those reducible via Riemann invariants, i.e. special coordinates of $u$ in which the system becomes diagonal) and for small, decaying initial data one even has global smooth solutions (the weak-linear-degeneracy theory of Li, Kong, and Zhou).

Finite-time breakdown can nevertheless occur for totally LD systems when the data are large: Jeffrey (1974) exhibited a strictly hyperbolic $3\times 3$ system with constant eigenvalues whose characteristics are all parallel, yet whose explicit $C^1$ solution blows up at finite time with $u_2=\ln(1-t/\alpha)\to-\infty$ — the solution amplitude itself diverges while $\partial_x u$ stays bounded. Such amplitude-driven breakdown, behaving like a blowing-up ordinary differential equation (compare Riccati's equation), is called a singularity of ODE type. Majda's 1984 monograph then in essence conjectured that for totally LD systems this is the only possible mechanism: such systems typically have smooth global solutions for any smooth initial data, unless $u(t,x)$ escapes every compact subset as $t$ approaches a finite $T_*$ — in particular, shock formation never happens. Brenier later phrased the same belief as: solutions of LD systems of conservation laws are expected to be smooth or to blow up in sup norm, not in Lipschitz norm. The conjecture is equivalent (as observed by Li's school) to a continuation criterion: a uniform a priori bound on $\|u(t,\cdot)\|_{L^\infty}$ on $[0,T]$ for every $T<T_0$ should imply a uniform bound on $\|\partial_x u(t,\cdot)\|_{L^\infty}$, with constants depending only on $T_0$ and the system; otherwise the $C^1$ solution could be continued, so breakdown requires amplitude blow-up.

The conjecture has been confirmed only for structured classes: diagonal systems, homogeneous or inhomogeneous (Li–Peng, Yang); $2$-step and, under additional hypotheses, $m$-step completely reducible systems (systems diagonalizable up to a triangular block coupling resolved step by step); block successive closed and partly rich LD systems of conservation laws (Li–Peng–Yang–Zhou); and totally LD systems with dissipative relaxation, including the Kerr–Debye model (a nonlinear-optics model coupling an electromagnetic field to a medium whose polarization response lags the field), which exhibit semilinear behavior (Carbou–Hanouzet–Natalini). Qu (2011) recast the conjecture for systems with source terms as an equivalent statement for compactly supported data: if $\|u\|_{L^\infty}$ is uniformly bounded on $[0,T]$, then $\|\partial_x u\|_{L^p}$ ($1<p\le\infty$) should be uniformly bounded; via an energy method in the $C^1$ framework he showed that whenever a singularity does form, all $W^{1,p}$ norms of the solution blow up simultaneously (a promotion principle lifting $L^p$-information on $\partial_x u$), verified the conjecture for the Kerr–Debye system and for blocked, completely reducible systems, and left the general case open. As Li's 2009 survey states plainly, for non-completely-reducible systems nothing is known — how to prove the conjecture in general, or how to construct a counterexample showing it fails, is open.

Problem Statement

Settle Majda's conjecture on the singularity mechanism for totally linearly degenerate quasilinear hyperbolic systems in one space dimension. Consider the Cauchy problem

$$\partial_t u+A(u)\,\partial_x u=F(u),\qquad u(0,x)=\varphi(x),$$

where $u$ takes values in $\mathbb{R}^n$, $A(u)$ is a smooth matrix with $n$ distinct real eigenvalues (strict hyperbolicity), every characteristic field satisfies Lax's linear degeneracy condition $\nabla\lambda_i(u)\cdot r_i(u)\equiv 0$ (where $\lambda_i(u)$ denotes the $i$-th eigenvalue of $A(u)$ and $r_i(u)$ a corresponding right eigenvector), $F(u)$ is a smooth source ($F\equiv 0$ is Majda's original conservation-law setting; $F\neq 0$ is the formulation of Qu's 2011 paper, covering models such as Kerr–Debye), and $\varphi$ is $C^1$ with bounded $C^1$ norm. Prove that if the maximal classical $C^1$ solution breaks down at a finite lifespan $T_*$, then the breakdown is of ODE type:

$$\|u(t,\cdot)\|_{L^\infty}\to\infty\quad\text{as }t\uparrow T_*,$$

so that no shock-type (gradient-only) breakdown can occur; equivalently, prove the continuation criterion that for compactly supported data a uniform $L^\infty$ bound on $u$ over $[0,T]$ for every $T<T_0$ implies a uniform $L^p$ bound ($1<p\le\infty$) on $\partial_x u$ over $[0,T]$, with constants depending only on the system and $T_0$. Or disprove the conjecture by exhibiting a strictly hyperbolic, totally linearly degenerate system within this class and admissible $C^1$ initial data of bounded $C^1$ norm whose $C^1$ solution breaks down in finite time while $\|u(t,\cdot)\|_{L^\infty}$ remains bounded. The answer should hold for arbitrary (large) initial data, not only small or decaying data.

References

1. Peng Qu, "Mechanism of singularity formation for quasilinear hyperbolic systems with linearly degenerate characteristic fields", Journal of Differential Equations 251(8):2066–2081, 2011. DOI: 10.1016/j.jde.2011.07.005, https://doi.org/10.1016/j.jde.2011.07.005
2. Andrew Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables", Applied Mathematical Sciences 53, Springer-Verlag, New York, 1984. ISBN 978-0-387-96037-1 (print) / 978-1-4612-1116-7 (online), DOI: 10.1007/978-1-4612-1116-7, https://link.springer.com/book/10.1007/978-1-4612-1116-7
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