For quasilinear wave equations in two and three space dimensions with smooth, compactly supported initial data of amplitude ε\varepsilon, the John–Hörmander theory predicts a sharp lifespan scale: Tε≈τˉ02/ε2T_\varepsilon\approx \bar\tau_0^{2}/\varepsilon^2 in 2D and Tε≈exp⁡(τ∗/ε)T_\varepsilon\approx\exp(\tau_*/\varepsilon) in 3D, where the constants τˉ0\bar\tau_0 and τ∗\tau_* are computed explicitly from the Friedlander radiation field of the data and the quadratic part of the nonlinearity. Hörmander's 1987 Lecture Notes in Mathematics 1256 chapter established the lower bound; the accompanying remark that the lower bound is expected to always give the precise asymptotic lifespan — with no proof except John's rotationally symmetric 3D case — is documented in Hörmander's 1997 Lectures on Nonlinear Hyperbolic Differential Equations as quoted by Alinhac (1999). Later literature has resolved major parts: Alinhac (1999) proved the conjecture in 2D under a non-degeneracy condition on the maximum of the radiation profile, and the 2-D blow-up theory was extended to general quasilinear equations by Li–Witt–Yin and Ding–Witt–Yin, still under non-degeneracy-type hypotheses; in 3D, sharp-lifespan and shock-formation theorems were proved under structural hypotheses (Alinhac's non-degeneracy; Christodoulou's and Christodoulou–Miao's compression conditions; Speck's near-spherical symmetry, with a sketched elimination of non-degeneracy; and, for 2-D multi-speed systems, shock formation near plane symmetry in Speck's 2018 Annals of PDE paper); and Klainerman–Wang–Yang–Yu (2026) proved the full unrestricted statement for the 3D irrotational compressible Euler equations. On the complementary side of the blow-up/global-existence borderline, Zha (2019) settled 2-D global and almost global existence in the null-condition regime. The surviving open core of the conjecture is precise and nonempty: the unrestricted all-data sharp-lifespan theorem for the general 3D quasilinear wave class, and removal of the non-degeneracy hypotheses in the sharp 2D statements. Adjacent open problems flagged by later literature — the full compressible Euler system with vorticity and entropy (announced as in preparation), multi-speed systems and their interaction zones, and continuation of solutions beyond the first shock — are recorded here as context rather than as part of the target.

Background

Nonlinear wave equations can develop singularities from arbitrarily smooth, compactly supported, arbitrarily small Cauchy data, in sharp contrast to the linear theory. For a quasilinear wave equation, that is, an equation of the form ∑α,β=0n(g−1)αβ(∂Φ) ∂α∂βΦ=0\sum_{\alpha,\beta=0}^{n}(g^{-1})^{\alpha\beta}(\partial\Phi)\,\partial_\alpha\partial_\beta\Phi=0 whose principal coefficients depend on the first derivatives ∂Φ\partial\Phi of the unknown and reduce to the Minkowski metric when ∂Φ=0\partial\Phi=0, the lifespan TεT_\varepsilon is the supremum of times up to which a smooth classical solution exists for data (εf,εg)(\varepsilon f,\varepsilon g), where f,gf,g are fixed smooth functions supported in a ball and ε>0\varepsilon>0 is a small amplitude. John (1974) proved finite-time breakdown in one space dimension, and John and Klainerman (1984) proved almost global existence in three space dimensions, Tε≳exp⁡(c/ε)T_\varepsilon\gtrsim\exp(c/\varepsilon); John also proved 3D blow-up results, with a matching upper bound in the rotationally symmetric case, as documented in the later works cited below. In that sharp radial case, and in Burgers' equation as model, the singularity is caused by the collapse of the separation between neighboring characteristics: first derivatives remain bounded while second derivatives blow up.

Hörmander, in the Lecture Notes in Mathematics 1256 chapter and in his book Lectures on Nonlinear Hyperbolic Differential Equations, systematized the sharp lower bound. The construction passes through the Friedlander radiation field: for data (f,g)(f,g), F0(ω,q)=14π(cˉ−1Rg(ω,q)−∂qRf(ω,q))F_0(\omega,q)=\frac{1}{4\pi}(\bar c^{-1}Rg(\omega,q)-\partial_q Rf(\omega,q)) with RR the Radon transform Rh(ω,q)=∫y⋅ω=qh(y) dSyRh(\omega,q)=\int_{y\cdot\omega=q}h(y)\,dS_y, cˉ\bar c the background sound speed (cˉ=1\bar c=1 under the normalization (g−1)αβ(0)=mαβ(g^{-1})^{\alpha\beta}(0)=m^{\alpha\beta}) and q=r−cˉ tq=r-\bar c\,t the retarded time variable, which gives the leading asymptotic profile of the solution of the linearized equation in the wave zone, the region ∣x∣≈t|x|\approx t where the decay of spherical waves accumulates. Approximating the nonlinear dynamics in the wave zone by a scalar asymptotic system of Burgers type for the outgoing profile — the near-simple-wave approximation, a simple wave being a solution carried by a single family of characteristics — Hörmander proved the lower bound lim inf⁡ε→0εlog⁡Tε≥τ∗\liminf_{\varepsilon\to 0}\varepsilon\log T_\varepsilon\ge\tau_*, where τ∗=(max⁡ω∈S2, q12G(ω) ∂q2F0(ω,q))−1\tau_*=(\max_{\omega\in S^2,\,q}\frac{1}{2}G(\omega)\,\partial_q^2F_0(\omega,q))^{-1} and G(ω)G(\omega) is a coefficient determined by the quadratic part of the nonlinearity in the outgoing null direction (explicitly computed from the quadratic terms of the equation; see Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Theorem 6.5.9, p. 129, and Klainerman–Wang–Yang–Yu 2026, Theorem 1.1, where G=−2cˉ−1℘G=-2\bar c^{-1}\wp for the Euler potential equation); under the genuine-nonlinearity (null-condition-violating) hypothesis imposed in the Problem Statement below, τ∗>0\tau_*>0 for nontrivial data — Klainerman–Wang–Yang–Yu (2026) record that τ∗\tau_* vanishes only for trivial data and that a genuine-nonlinearity-type condition is needed to exclude null-condition equations — and the maximizing point (ω∗,q∗)(\omega^*,q^*) identifies the direction and retarded time at which the first singularity is predicted. In two space dimensions the corresponding scale is algebraic, lim inf⁡ε→0εTε1/2≥τˉ0\liminf_{\varepsilon\to 0}\varepsilon T_\varepsilon^{1/2}\ge\bar\tau_0, because spherical waves decay more slowly.

The sharp-lifespan question — whether these lower bounds are in fact equalities, with blow-up of John's type for all nontrivial small data without symmetry assumptions — is known as the John–Hörmander conjecture. It concerns the genuinely nonlinear regime, opposite to the null condition regime: when the quadratic nonlinearity vanishes on null pairs, that is, on pairs of Minkowski-orthogonal null covectors (the null condition), small-data solutions exist globally in three dimensions, and the sharp-lifespan problem sits exactly on this blow-up/global-existence borderline. The modern approach to the upper bound is geometric: one endows spacetime with the acoustical metric g(∂Φ)g(\partial\Phi) associated with the equation, foliates the exterior of the wave zone by level sets of an eikonal function uu (solving gαβ∂αu∂βu=0g^{\alpha\beta}\partial_\alpha u\partial_\beta u=0), and tracks the inverse foliation density μ\mu (the Jacobian measuring the separation of neighboring null hypersurfaces); shock formation is equivalent to μ→0\mu\to 0, that is, to the intersection of characteristics. Christodoulou's monograph (2007) and Christodoulou–Miao's book (2014) developed this framework for compressible fluid systems under compression hypotheses; Speck's monograph (2016) proved the sharp classical lifespan for 3D quasilinear wave equations with nearly spherically symmetric data, and his 2018 Annals of PDE paper proved shock formation for 2-D multi-speed quasilinear systems — systems whose several characteristic families propagate with distinct wave speeds — for perturbations of plane-symmetric simple waves, with non-trivial wave interactions persisting up to the singularity. For the compressible Euler equations of gas dynamics, irrotational (vanishing curl of the velocity) isentropic (constant-entropy) flow reduces to a single quasilinear wave equation for a velocity potential φ\varphi, so Euler is a central special case; the full system, with vorticity (nonzero curl of the fluid velocity) and non-constant entropy, does not reduce to a scalar equation, and its unrestricted sharp-lifespan behavior remains the outstanding case (Klainerman–Wang–Yang–Yu 2026 settle the irrotational case and state the full case as work in progress).

Problem Statement

Consider the Cauchy problem for a quasilinear wave equation in three space dimensions, ∑α,β=03(g−1)αβ(∂Φ) ∂α∂βΦ=0,(g−1)αβ(0)=mαβ,\sum_{\alpha,\beta=0}^{3}(g^{-1})^{\alpha\beta}(\partial\Phi)\,\partial_\alpha\partial_\beta\Phi=0,\qquad (g^{-1})^{\alpha\beta}(0)=m^{\alpha\beta}, with smooth initial data Φ∣t=0=εf\Phi|_{t=0}=\varepsilon f, ∂tΦ∣t=0=εg\partial_t\Phi|_{t=0}=\varepsilon g, where f,gf,g are fixed smooth functions, compactly supported, and not both trivial, the quadratic part of the nonlinearity violates the null condition (genuine nonlinearity in the outgoing direction), and TεT_\varepsilon is the lifespan of the classical solution. Let F0F_0 be the Friedlander radiation field of (f,g)(f,g), RR the Radon transform, q=r−cˉ tq=r-\bar c\,t the retarded time variable (cˉ\bar c the background sound speed; cˉ=1\bar c=1 under the normalization (g−1)αβ(0)=mαβ(g^{-1})^{\alpha\beta}(0)=m^{\alpha\beta} above), G(ω)G(\omega) the coefficient determined by the outgoing quadratic nonlinearity (explicitly computed from the quadratic terms of the equation; see Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Theorem 6.5.9, p. 129, and Klainerman–Wang–Yang–Yu 2026, Theorem 1.1), and τ∗=(max⁡ω∈S2, q12G(ω)∂q2F0(ω,q))−1>0.\tau_*=\Big(\max_{\omega\in S^2,\,q}\tfrac{1}{2}G(\omega)\partial_q^2F_0(\omega,q)\Big)^{-1}>0. Determine whether Hörmander's lower bound always gives the precise asymptotic lifespan of the solutions and the breakdown mechanism is universal, specifically: (1) Is lim⁡ε→0εlog⁡Tε=τ∗\lim_{\varepsilon\to 0}\varepsilon\log T_\varepsilon=\tau_*, and does classical regularity necessarily break down by shock formation — vanishing of the inverse foliation density of an acoustical eikonal foliation, equivalently the crossing of neighboring characteristics, with ∂Φ\partial\Phi bounded and second derivatives blowing up — at the boundary of the maximal Cauchy development (the largest region on which the classical solution is uniquely determined by the data), for every nontrivial compactly supported datum, with no assumptions on the data beyond smallness (in particular no symmetry, near-radiality, compression, or non-degeneracy hypotheses), the equation class being the genuinely nonlinear class fixed above? Here the shock time TεeT_\varepsilon^e at the boundary of the maximal Cauchy development is distinguished from the classical lifespan TεT_\varepsilon and satisfies Tεe≥TεT_\varepsilon^e\ge T_\varepsilon; in the settled irrotational-Euler case the two limits coincide, lim⁡ε→0εlog⁡Tε=lim⁡ε→0εlog⁡Tεe=τ∗\lim_{\varepsilon\to 0}\varepsilon\log T_\varepsilon=\lim_{\varepsilon\to 0}\varepsilon\log T_\varepsilon^e=\tau_* (Klainerman–Wang–Yang–Yu 2026, Theorem 1.2). (2) In two space dimensions, where the predicted scale is lim⁡ε→0εTε1/2=τˉ0\lim_{\varepsilon\to 0}\varepsilon T_\varepsilon^{1/2}=\bar\tau_0 with τˉ0=(max⁡(σ,ω)∈R×S1∂σ2R(1)(σ,ω))−1\bar\tau_0=(\max_{(\sigma,\omega)\in\mathbb{R}\times S^1}\partial_\sigma^2R^{(1)}(\sigma,\omega))^{-1}, R(1)R^{(1)} being Alinhac's first radiation profile of the two-dimensional data — the outgoing asymptotic profile of the linearized solution, given by a fractional integral in the retarded time of the Radon transform of the data (Alinhac 1999, eq. (0.3); in his notation R(1)(σ,ω)=122π∫s≥σ1s−σ[R(s,ω,u11)−∂sR(s,ω,u01)] dsR^{(1)}(\sigma,\omega)=\frac{1}{2\sqrt{2\pi}}\int_{s\ge\sigma}\frac{1}{\sqrt{s-\sigma}}[R(s,\omega,u_{11})-\partial_sR(s,\omega,u_{01})]\,ds with u01=fu_{01}=f, u11=gu_{11}=g the leading data profiles, that is, R(1)(σ,ω)=122π∫s≥σ1s−σ[R(s,ω,g)−∂sR(s,ω,f)] ds,R^{(1)}(\sigma,\omega)=\frac{1}{2\sqrt{2\pi}}\int_{s\ge\sigma}\frac{1}{\sqrt{s-\sigma}}\big[R(s,\omega,g)-\partial_sR(s,\omega,f)\big]\,ds, where ω∈S1\omega\in S^1 is the outgoing direction, σ\sigma the retarded time and R(s,ω,⋅)R(s,\omega,\cdot) the Radon transform ∫x⋅ω=s(⋅) dx\int_{x\cdot\omega=s}(\cdot)\,dx): does the sharp statement hold for all nontrivial data whose ∂σ2R(1)\partial_\sigma^2R^{(1)} attains a positive maximum (so that τˉ0>0\bar\tau_0>0), once the non-degeneracy hypothesis — that this positive maximum is attained at a single non-degenerate point, which is Alinhac's own 1999 hypothesis (ND) rather than part of Hörmander's conjecture — is removed?

References

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