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

For the Cauchy problem of a one-dimensional first-order quasilinear hyperbolic system $\partial_t u+A(u)\partial_x u=F(u)$ with $C^2$ coefficients, strict hyperbolicity, and all characteristic fields linearly degenerate in the sense of Lax, a conjecture attributed to A. Majda (1984) asserts that any finite-time breakdown of the classical $C^1$ solution must be of ODE type: the amplitude $\|u(t,\cdot)\|_{L^\infty}$ itself blows up at the breakdown time, and no shock-type singularity (gradient blow-up with uniformly bounded amplitude) can occur. Peng Qu (2011) proposed an a priori estimate (his Conjecture 2) that he asserts to be equivalent to the conjecture: for compactly supported $C^1$ data, a uniform $L^\infty$ bound on the solution forces a uniform $L^p$ bound on $\partial_x u$ for some $1\le p\le\infty$, hence a uniform $C^1$ bound and continuation of the solution. Qu proved a promotion principle under a structural noninteraction condition (his condition (32)): all $W^{1,p}$ norms ($1\le p\le\infty$) of a $C^1$ solution blow up simultaneously, giving a method to verify the ODE-type property that applies in particular to generalized Kerr–Debye relaxation models; the prior literature verified the conjecture only for diagonal, reducible, and block-structured systems. The conjecture remains open for the general class, and no counterexample is known. The problem asks for a proof or a refutation of Majda's conjecture for arbitrary systems of the stated class.

Background

A one-dimensional first-order quasilinear hyperbolic system has the form

$$\partial_t u+A(u)\,\partial_x u=F(u),\qquad u:\mathbb{R}_t\times\mathbb{R}_x\to\mathbb{R}^n,$$

where $A(u)$ is an $n\times n$ matrix depending smoothly on the state $u$ and $F(u)$ is a smooth source term. The system is called hyperbolic if $A(u)$ has real eigenvalues $\lambda_1(u),\dots,\lambda_n(u)$ (the characteristic speeds) with a full set of right eigenvectors $r_i(u)$, and strictly hyperbolic if the eigenvalues are pairwise distinct. Following Lax, the $i$-th characteristic field is genuinely nonlinear if $\nabla\lambda_i(u)\cdot r_i(u)\neq 0$ (the speed changes along the wave, which is the mechanism of compressive shock formation), and linearly degenerate if $\nabla\lambda_i(u)\cdot r_i(u)\equiv 0$ (the speed is constant along the wave, as for contact discontinuities in gas dynamics). For genuinely nonlinear scalar laws and systems, gradient blow-up of classical solutions is generic (Lax; John 1974). When all characteristic fields are linearly degenerate, the direct Lax compression mechanism is absent, and for small, suitably decaying data global classical solutions exist (the weak linear degeneracy theory of Zhou 2004 builds on this). Nevertheless, breakdown of $C^1$ solutions in finite time can still occur for general data even for completely exceptional (i.e. linearly degenerate) systems, as observed already by Jeffrey (1974). The question then arises which mechanism drives this breakdown. A singularity is called ODE-type or semilinear if the amplitude $\|u(t,\cdot)\|_{L^\infty}$ itself blows up, governed roughly by the underlying ordinary differential equation $u'=F(u)$; it is called shock-type or gradient-type if the amplitude stays bounded while some spatial derivative $\partial_x u$ blows up. Majda's 1984 book conjectured that for strictly hyperbolic one-dimensional quasilinear systems with all characteristic fields linearly degenerate, every finite-time blow-up of a $C^1$ solution must be of ODE type; equivalently, a uniform bound on the amplitude implies global existence of the classical solution. Partial verifications before 2011 all imposed additional structure: reducible systems, meaning systems that can be transformed into diagonal or block-diagonal form by a change of variables (Li and Peng 2003, who also gave an example showing the $C^0$ norm can indeed blow up), diagonal systems (Yang 2009 for the Cauchy problem, Yang 2011 for a Goursat problem, i.e. a characteristic initial-boundary value problem with data prescribed on transversal characteristic curves), block-structured systems with strict block hyperbolicity, part richness, and block-closedness (Li, Peng, Yang, and Zhou 2008), blocked completely reducible systems without part richness (Qu 2011), and totally linearly degenerate relaxation systems such as Kerr–Debye models, relaxation models for nonlinear optics in which the medium's refractive index responds with a delay to the light intensity, via energy methods at higher ($H^2$) regularity (Carbou, Hanouzet, and Natalini 2009). Peng Qu's 2011 paper introduced an energy method working directly in the $C^1$ framework and proved a promotion principle: under a structural noninteraction condition (his condition (32), which allows certain non-strictly-hyperbolic block systems and covers generalized Kerr–Debye relaxation models), all $W^{1,p}$ norms ($1\le p\le\infty$) of a $C^1$ solution with compactly supported, bounded-$C^1$ initial data blow up simultaneously, giving a way to verify the ODE-type property where an amplitude-to-gradient estimate is independently available. Crucially, he proposed an a priori estimate, his Conjecture 2, which he asserts to be equivalent to the conjecture via this promotion principle: a uniform $L^\infty$ bound on $u$ (with compactly supported data) forces a uniform $L^p$ bound on $\partial_x u$, which by the promotion principle would yield a uniform $C^1$ bound and hence continuation by the standard local existence theory. What the paper does not do is establish Conjecture 2, and hence Majda's conjecture, for systems satisfying no structural condition beyond strict hyperbolicity and linear degeneracy of all fields.

Problem Statement

Consider the Cauchy problem for a one-dimensional first-order quasilinear hyperbolic system

$$\partial_t u+A(u)\,\partial_x u=F(u),\qquad u(0,x)=u_0(x),\qquad u\in\mathbb{R}^n,$$

where $A$ and $F$ are $C^2$, $A(u)$ has $n$ pairwise distinct real eigenvalues $\lambda_1(u),\dots,\lambda_n(u)$ with corresponding right eigenvectors $r_i(u)$ for all relevant $u$ (strict hyperbolicity), every characteristic field is linearly degenerate in the sense of Lax, $\nabla\lambda_i(u)\cdot r_i(u)\equiv 0$ for every $i$, and the initial datum $u_0$ is $C^1$ with bounded $C^1$ norm and compact support. Prove or disprove Majda's conjecture for this class in full generality: if the maximal classical $C^1$ solution breaks down at a finite time $T_*$, then

$$\lim_{t\uparrow T_*}\|u(t,\cdot)\|_{L^\infty(\mathbb{R})}=+\infty,$$

that is, every finite-time singularity is of ODE type and no shock-type (gradient-only) blow-up with uniformly bounded amplitude can occur. A refutation consists of an explicit system in the class and admissible initial data for which the $C^1$ solution breaks down in finite time while $\|u(t,\cdot)\|_{L^\infty}$ remains uniformly bounded. Qu (2011) proposes an a priori estimate (his Conjecture 2) — that a uniform $L^\infty$ bound on the solution over $[0,T]$, together with the stated hypotheses on the data, forces a uniform $L^p$ bound on $\partial_x u$ over $[0,T]$ for some $p$ with $1\le p\le\infty$, and hence, via the promotion principle and the standard local existence theory, a uniform $C^1$ bound and continuation of the solution — and asserts it to be equivalent to the conjecture. The conjecture is posed here for the stated class: strictly hyperbolic systems with $C^2$ coefficients, all characteristic fields linearly degenerate in the sense of Lax, and compactly supported $C^1$ initial data.

References

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