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

For one-dimensional first-order quasilinear hyperbolic systems $u_t+A(u)u_x=F(u)$ that are strictly hyperbolic and whose characteristic fields are all linearly degenerate in the sense of Lax, Majda conjectured (1984) that any finite-time blow-up of a classical $C^1$ solution must be of ODE type: the amplitude $\|u(t,\cdot)\|_{L^\infty}$ itself blows up at the first singular time. All known confirmations are restricted to special structures (diagonal, rich or part-rich, block-structured, Kerr-Debye, Chaplygin gas, specific MHD systems) or are conditional on a priori amplitude and derivative bounds. The problem is to prove the conjecture for general systems of this class, with no block-structure or a priori-bounds hypotheses, or to construct a counterexample in which derivatives blow up while the solution amplitude stays bounded.

Background

A one-dimensional first-order quasilinear hyperbolic system is a Cauchy problem

$$u_t+A(u)u_x=F(u),\qquad u(0,x)=u_0(x),$$

where $u:\mathbb{R}_t\times\mathbb{R}_x\to\mathbb{R}^n$, the coefficient matrix $A(u)$ is $C^2$ and has $n$ real eigenvalues $\lambda_1(u)<\cdots<\lambda_n(u)$ with right eigenvectors $r_j(u)$ (strict hyperbolicity), and $F$ is a $C^1$ source term. A characteristic field $j$ is called genuinely nonlinear in the sense of Lax when $\nabla\lambda_j(u)\cdot r_j(u)\neq 0$, and linearly degenerate when $\nabla\lambda_j(u)\cdot r_j(u)\equiv 0$, i.e. the eigenvalue is constant along the integral curves of its own eigenvector field. Classical theory (Lax, John, Li) shows that genuine nonlinearity drives shock-type breakdown: first spatial derivatives blow up while the solution amplitude stays bounded, because characteristics of the same family cross. For linearly degenerate fields this compression mechanism is absent — the contact-discontinuity field of gas dynamics is the standard example — so finite-time blow-up, when it occurs, is expected to come from the coupling of the PDE with the ordinary differential equation $\dot u=F(u)$ along characteristics. A blow-up at time $T_*$ is called an ODE-type singularity if $\limsup_{t\to T_*^{-}}\|u(t,\cdot)\|_{L^\infty}=+\infty$, i.e. the amplitude itself escapes every bound. The study of systems whose fields are all linearly degenerate (Jeffrey's completely exceptional systems) goes back to the 1970s, and in his 1984 monograph Majda posed the conjecture that for strictly hyperbolic systems with all characteristic fields linearly degenerate and $C^1$ initial data of bounded $C^1$ norm, every finite-time blow-up of the classical solution is necessarily of ODE type. The conjecture is equivalently formulated by Qu as a continuation principle: for compactly supported initial data, if $\|u(t,\cdot)\|_{L^\infty}$ is uniformly bounded on $[0,T]$, then so is $\|u(t,\cdot)\|_{W^{1,p}}$ for every $1<p\le\infty$, which opens the energy method to the problem. Partial confirmations cover diagonal and inhomogeneous diagonal systems (Li–Peng; Yang), rich and part-rich systems — classes singled out by differential identities among the eigenvalue derivatives of $A(u)$ that supply the cancellations needed in the characteristic estimates — and successively block-closed or completely reducible architectures (Li–Peng–Yang–Zhou, whose part-richness identity is of exactly this kind), the Kerr–Debye system of nonlinear optics, a relaxation model coupling the electromagnetic field to a polarization variable with finite response time (Carbou–Hanouzet–Natalini; Qu), blocked systems — in which the unknown $u$ is split into blocks and $A(u)$ is block-triangular, so that the coupling between blocks is one-directional — without part-richness but under uniform a priori bounds on $\|u\|_{L^\infty}$ and $\|\partial_xu^{(1)}\|_{L^\infty}$ (Qu), periodic solutions (Qu), and individual physical models such as the Chaplygin gas, an isentropic flow with the exotic equation of state $p=-k/\rho$ (Kong–Wei–Zhang), and a four-field nonhomogeneous MHD system (Hu–Zeng), in all of which the blow-up amplitude indeed diverges. What no argument so far achieves is the unrestricted statement: the coupling between several linearly degenerate fields through the full matrix $A(u)$ is not controlled by any known estimate without structural hypotheses or assumed a priori bounds.

Problem Statement

Prove or disprove Majda's conjecture for one-dimensional quasilinear hyperbolic systems. Let $u_t+A(u)u_x=F(u)$, $u:\mathbb{R}_t\times\mathbb{R}_x\to\mathbb{R}^n$, with $A\in C^2$ strictly hyperbolic (real eigenvalues $\lambda_1(u)<\cdots<\lambda_n(u)$) and all characteristic fields linearly degenerate, $\nabla\lambda_j(u)\cdot r_j(u)\equiv 0$ for $j=1,\dots,n$, let $F\in C^1$, and let $u_0$ be $C^1$ with bounded $C^1$ norm. Suppose the classical $C^1$ solution exists on $[0,T_*)$ and blows up at the finite time $T_*$. Is it necessarily true that

$$\limsup_{t\to T_*^{-}}\|u(t,\cdot)\|_{L^\infty}=+\infty,$$

i.e. that the singularity is of ODE type rather than a derivative-only (shock-type) blow-up? Equivalently, in Qu's continuation-principle form (stated for compactly supported initial data): does a uniform bound on $\|u(t,\cdot)\|_{L^\infty}$ over $[0,T]$ imply a uniform bound on $\|u(t,\cdot)\|_{W^{1,p}}$ for every $1<p\le\infty$? A complete answer must hold for arbitrary systems of this class — without block-structure assumptions such as blocked form, strict block-hyperbolicity, complete reducibility or part-richness, without constant-multiplicity restrictions, and without a priori bounds on $\|u\|_{L^\infty}$ or on any block's spatial derivative being assumed. A disproof consists of an explicit system in this class and initial data for which the $C^1$ solution blows up at finite time while $\|u(t,\cdot)\|_{L^\infty}$ remains bounded.

References

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