Sharp structural characterization of small-data global classical solvability for one-dimensional quasilinear hyperbolic systems with source terms

For the one-dimensional Cauchy problem $\partial_t u + A(u)\partial_x u = F(u)$ with strictly hyperbolic $A(u)$, equilibrium source $F(0)=0$, and sufficiently small, spatially decaying $C^1$ initial data, the known theory gives only sufficient conditions for existence of a global-in-time classical (C^1) solution: either all characteristic fields are strictly damped by the source, or (Liu–Qu's partially dissipative regime) some fields are strictly damped while the remaining ones are weakly linearly degenerate and the source vanishes along their characteristic trajectories. The homogeneous case $F\equiv 0$ is completely classified by weak linear degeneracy, but with a general source only fragments of the necessity side are known (a single 2×2 blow-up counterexample and breakdown theorems under an additional matching condition). The open problem is the missing sharp characterization: determine necessary and sufficient structural conditions on the pair $(A,F)$ under which the small-data Cauchy problem admits a unique global classical solution for every admissible small datum, so that no derivative blow-up (shock formation) occurs, and establish the matching blow-up constructions outside the characterized class.

Background

A first-order quasilinear hyperbolic system in one space dimension is $\partial_t u + A(u)\partial_x u = F(u)$, where $u(t,x)\in\mathbb{R}^n$, $A(u)$ is a smooth $n\times n$ matrix, and $F(u)$ is a smooth source term with equilibrium $F(0)=0$. The system is strictly hyperbolic near $u=0$ if $A(u)$ has $n$ distinct real eigenvalues $\lambda_1(u)<\cdots<\lambda_n(u)$, with smooth left and right eigenvectors $l_i(u)$, $r_i(u)$ normalized by $l_i r_j=\delta_{ij}$. The $i$-th characteristic trajectory through the origin is the integral curve $u=u^{(i)}(s)$ of the right eigenvector field, $du^{(i)}/ds=r_i(u^{(i)})$, $u^{(i)}(0)=0$. A characteristic speed $\lambda_i$ is linearly degenerate in the sense of Lax if $\nabla\lambda_i\cdot r_i\equiv 0$ everywhere, and genuinely nonlinear if $\nabla\lambda_i\cdot r_i$ never vanishes; genuine nonlinearity drives the nonlinear steepening of waves whose end state is shock formation, i.e. finite-time blow-up of $u_x$ (breakdown of the classical solution) even for arbitrarily small and smooth data. Li Ta-tsien's school introduced the weaker notion of weak linear degeneracy (WLD): $\lambda_i$ is weakly linearly degenerate if $\lambda_i(u^{(i)}(s))\equiv\lambda_i(0)$ along the $i$-th characteristic trajectory through the equilibrium, i.e. the speed is constant to all orders along that one curve but may vary elsewhere. For the homogeneous system ($F\equiv 0$) this notion yields a complete small-data dichotomy: Li, Zhou and Kong proved that WLD of all fields implies a unique global classical solution for all sufficiently small, decaying $C^1$ data, while Li–Kong and Kong proved sharp life-span asymptotics and finite-time breakdown for suitable small data whenever some field is not weakly linearly degenerate, so WLD is exactly the condition that rules out derivative blow-up for all small data.

With a source term present, the linearized coupling of $F$ to the characteristic fields is encoded by the matrix $G(u)=L(u)\nabla F(u)R(u)$, where $L$ and $R$ collect the left and right eigenvectors; a set of characteristic families indexed by a subset $\mathscr{P}$ is strictly dissipative when the source strictly damps those families, in the sense of the strict-dissipation hypothesis used in the corresponding papers (Zhou 2011; Liu–Qu 2012; Qu 2013), which produces exponential-in-time damping of these families' amplitudes. This is related to, but weaker than, the Shizuta–Kawashima condition — the standard algebraic hypothesis (coupling of the dissipative matrix to all characteristic directions) under which the Kawashima school constructs global smooth small-data solutions of hyperbolic balance laws. Two further structural conditions organize the inhomogeneous theory: the matching condition $\nabla F(u) r_i(u)\equiv 0$, under which the source is inertial along each characteristic family and the homogeneous blow-up mechanism survives, and the F-degeneracy condition $F(u^{(j)}(s))\equiv 0$ along the trajectory of each weakly linearly degenerate field. Liu and Qu proved a global existence theorem in the partially dissipative regime — the index set split into a strictly dissipative part $\mathscr{P}$ and a complementary part $\mathscr{Q}$ whose fields are weakly linearly degenerate and on whose trajectories the source vanishes, plus wave-interaction hypotheses later weakened by Qu — and gave a 2×2 counterexample showing that the F-degeneracy condition cannot simply be dropped. Qu–Liu and Qu–Wang extended these sufficient conditions (one weakly linearly degenerate characteristic; multi-dimensional systems violating the Kawashima condition), and Zhou (2011) gave a different proof of small-data global existence for weakly dissipative systems in the Hanouzet–Natalini setting. On the negative side, Li Shumin and Xu–Wang established breakdown theorems for non-WLD inhomogeneous systems, but only under the matching condition. Thus, in contrast with the homogeneous case, no necessary-and-sufficient structural characterization of small-data global classical solvability is known for $F\not\equiv 0$: it is open whether weak linear degeneracy of the undamped families is necessary once the matching condition fails, whether a nonlinear source can regularize a genuinely nonlinear undamped family, and whether global solvability holds outside the partially dissipative interaction subclass.

Problem Statement

Consider the Cauchy problem for the one-dimensional strictly hyperbolic system $\partial_t u + A(u)\,\partial_x u = F(u),\qquad u(0,x)=\varphi(x),$ where $A$ and $F$ are sufficiently smooth (say $C^\infty$, as in the Background) near $u=0$, $A(u)$ has $n$ distinct real eigenvalues near $0$, $F(0)=0$, and the admissible data are $C^1$ functions small in a weighted norm, e.g. $\theta=\sup_x (1+|x|)^{1+\mu}\,(|\varphi|+|\varphi'|)\le\theta_0$ for fixed $\mu>0$ and some $\theta_0>0$. Determine sharp necessary and sufficient structural conditions on the pair $(A,F)$, defined in a neighborhood of the equilibrium, under which the Cauchy problem admits a unique global-in-time classical solution $u\in C^1([0,\infty)\times\mathbb{R})$ for every admissible small datum — equivalently, under which no admissible small datum develops derivative blow-up (shock formation). Such a characterization consists of global existence under the stated conditions together with the converse — for every system violating the conditions, an admissible arbitrarily small datum whose classical solution breaks down in finite time — and the known classes defined in the corresponding papers arise as special cases: the homogeneous weakly linearly degenerate systems of Li–Zhou–Kong, the strictly dissipative systems, and Liu–Qu's partially dissipative systems, in which a part of the equations possesses strict dissipation while the remaining characteristic fields are weakly linearly degenerate with the source vanishing along their characteristic trajectories, under the wave-interaction hypotheses. In particular, such a characterization would settle, as consequences of the main goal, the currently open necessity questions: whether weak linear degeneracy of non-dissipative characteristic families is necessary when the matching condition $\nabla F\,r_i\equiv 0$ fails, whether the source-vanishing condition $F(u^{(j)}(s))\equiv 0$ along weakly linearly degenerate trajectories is necessary in general (beyond Liu–Qu's 2×2 example), and whether any system outside the partially dissipative interaction subclass is globally solvable for all small data.

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