The weak null conjecture for systems of nonlinear wave equations

The weak null conjecture asks whether the weak null condition — global solvability, with at most exponential growth in the logarithmic time $s=\varepsilon\log t$, of the asymptotic system obtained by a geometric-optics reduction near the outgoing light cone — is sufficient for global regularity of the Cauchy problem for systems of nonlinear wave equations in $3+1$ dimensions with small, smooth, compactly supported data. Stated in this generality by Deng and Pusateri (2020) and standardly attributed to Lindblad and Rodnianski's work on the Einstein equations in harmonic gauge, the conjecture was disproved in its unrestricted form by Kadar (2025): a weak-null system with undifferentiated quadratic semilinear terms can blow up for arbitrarily small data. The derived restricted conjecture — the same statement for systems whose quadratic semilinear terms are all products of first derivatives of the fields — remains open; the known partial results (hierarchical systems, bounded stable asymptotic systems) all exclude the borderline exponential-growth regime that the quasilinear scalar prototype already exhibits. This record asks for a proof or a refutation of the restricted weak null conjecture, covering coupled systems.

Background

The small-data global regularity problem for nonlinear wave equations asks: given a system of wave equations in three space dimensions (spacetime dimension $3+1$) with smooth initial data — the Cauchy problem, prescribing the unknowns and their time derivatives at $t=0$ — that are compactly supported (vanishing outside a ball) and of size $\varepsilon\ll1$, does the solution stay smooth and bounded for all $t\ge0$? The answer is governed by the structure of the nonlinear terms. John (1981) proved that for broad classes of equations with quadratic nonlinearities, for example $\Box u=u_t^2$ where $\Box=-\partial_t^2+\Delta$ is the Minkowski wave operator, smooth solutions with arbitrarily small compactly supported data develop singularities in finite time (blowup). Christodoulou (1986) identified the null condition as a sufficient structural condition for global regularity: for a semilinear system $\Box u^I=\sum a^{I,\alpha\beta}_{JK}\,\partial_\alpha u^J\,\partial_\beta u^K$ (indices $\alpha,\beta$ run over $0,1,2,3$), the quadratic form $a^{I,\alpha\beta}_{JK}\xi_\alpha\xi_\beta$ must vanish on every null covector $\xi=(1,\omega)$ with $\omega\in S^2$, that is, on the directions tangent to the light cone. Under the null condition, small data yield global solutions that asymptotically behave like solutions of the linear wave equation.

The null condition is sufficient but not necessary. The Einstein vacuum equations of general relativity, written in wave (harmonic) gauge — coordinates $x^\mu$ chosen to satisfy the wave equation $\Box_g x^\mu=0$ with respect to the unknown metric $g$, under which the Einstein equations become a system of nonlinear wave equations — violate the null condition, yet Lindblad and Rodnianski (2003, 2010) proved that Minkowski spacetime is globally nonlinearly stable in this gauge. The structure that survives is captured by the weak null condition (WNC), defined through the asymptotic system: inserting the geometric-optics ansatz $u(t,x)\approx\varepsilon r^{-1}U(s,q,\omega)$, with $r=|x|$, retarded time $q=r-t$, direction $\omega=x/r$, and logarithmic time $s=\varepsilon\log t$, into the equation and keeping only the resonant quadratic interactions tangent to the outgoing light cone $r\approx t$ formally yields a reduced equation for the profile $U$; for null-condition systems this reduced equation is trivial ($\partial_s\partial_qU=0$). The system satisfies the WNC when this asymptotic system admits a global solution for all $s\ge0$, with $U$ and its derivatives growing at most like $e^{Cs}$ for some constant $C$ — that is, at most like $t^{C\varepsilon}$ in physical variables. The canonical quasilinear example is the scalar equation $-\partial_t^2u+(1+u)\Delta u=0$, whose asymptotic system is $\partial_s\partial_qU+U\,\partial_q^2U=0$; Lindblad (2008) proved its small-data global existence, the solutions growing like $t^{C\varepsilon}$ rather than scattering to linear waves. (Quasilinear means that the coefficients of the highest, second derivatives depend on the unknown or its first derivatives; semilinear terms are the nonlinear terms free of second derivatives.)

Deng and Pusateri (2020) studied the scalar class $g^{\alpha\beta}(u)\,\partial_\alpha\partial_\beta u=0$ under the WNC: they rigorously derived the asymptotic system, proved global existence for small compactly supported data, and proved modified scattering — convergence, after a nonlinear correction, to a solution of the linear wave equation. Their bounds come from commuting the equation with the Lorentz vector fields (the rotations, boosts, and scaling field generating the symmetries of $\Box$), and these derivatives grow at most like $e^{Cs}$; the paper leaves open whether uniform weighted energy estimates — $L^2$ bounds with radial weights encoding decay near the light cone — can be closed at this borderline growth, and states that its techniques should extend to general systems satisfying the WNC. They posed, as their Conjecture 1.1, that the WNC is a sufficient condition for global regularity of the Cauchy problem with small and localized data, noting that the conjecture is naturally attributed to Lindblad and Rodnianski, who had themselves only remarked that it was unclear whether their criterion suffices. Keir (2018) stated a precise version and proved it for hierarchical systems (each component's asymptotic equation involves only components that decay faster) via the p-weighted energy method, which weights the energy integrals by powers of the affine parameter along outgoing null cones. Kadar (2025) then disproved the conjecture in its unrestricted form: a system of his with undifferentiated quadratic semilinear terms (terms such as $\phi_1^2$ containing the fields themselves, not their derivatives) satisfies the WNC but blows up for arbitrarily small data. The derived open problem is the restricted weak null conjecture — the same statement for systems whose semilinear terms are all differentiated, i.e. products of first derivatives of the fields — which Yu (2025) reports as still open. For the scalar prototype, the companion question of describing the leading asymptotics of $u$ itself inside the light cone has since been completed (Yu 2021, 2024, 2025; Luk, Oh, and Yu 2025).

Problem Statement

The problem is the weak null conjecture in its surviving restricted form; the original unrestricted conjecture — that the weak null condition alone suffices for small-data global regularity — has been disproved (Kadar 2025; see Background), so the open question is the derived restricted variant, which is a strictly weaker claim than the original statement.

Precisely, consider systems of nonlinear wave equations on $\mathbb{R}^{1+3}$ with quadratic nonlinearities,

$(m^{\alpha\beta}+g^{\alpha\beta}(u,\partial u))\,\partial_\alpha\partial_\beta u^I=F^I(u,\partial u),\qquad I=1,\dots,N,$

with $m=\mathrm{diag}(-1,1,1,1)$, smooth coefficients $g^{\alpha\beta}$ vanishing at the origin, and a hyperbolic principal part (the second-derivative operator) for small data. The class restriction is that every quadratic semilinear term in $F$ — every term not involving second derivatives of the unknowns — is a product of first derivatives $\partial u^J\,\partial u^K$; undifferentiated factors such as $u^Ju^K$ are excluded, while the quasilinear coefficients $g^{\alpha\beta}(u,\partial u)$ may depend on both the fields and their first derivatives. Define the associated asymptotic system as the reduced equation for the profile $U$ obtained from the ansatz $u\approx\varepsilon r^{-1}U(s,q,\omega)$, with $s=\varepsilon\log t$, $q=r-t$, $\omega=x/r$, by retaining the resonant quadratic interactions, and say the system satisfies the weak null condition if, for all sufficiently small smooth decaying data, the asymptotic system admits a global solution for all $s\ge0$ whose derivatives grow at most like $e^{Cs}$ for some constant $C$.

Claim to be resolved: under these hypotheses, every sufficiently small, smooth, compactly supported Cauchy datum for the full system admits a unique global smooth solution (positive resolution); or, contrarily, there exists a system in this class satisfying the weak null condition as above for which arbitrarily small smooth compactly supported data lead to finite-time blowup or loss of regularity (negative resolution).

A positive resolution must hold for coupled multi-component systems, not only scalar equations. A negative resolution must verify the weak null condition for the exhibited system, that is, prove global solvability of its asymptotic system with the growth bound. The completion boundary is a proof or counterexample for the claim as stated; the leading-order asymptotics of $u$ inside the light cone is a separate objective, already complete for the scalar prototype class, and lies outside this boundary.

References

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