For the three-dimensional isentropic compressible Euler equations with the Chaplygin-gas pressure law p(ρ)=P0−B/ρp(\rho)=P_0-B/\rho, the system is totally linearly degenerate, and Majda's 1984 conjecture predicts that such multidimensional systems should admit global smooth solutions for small Sobolev data. This is known only in special classes: irrotational data (where the system reduces to a null-form wave equation), axisymmetric data in 2D, and short-pulse data in 2D. For genuinely non-irrotational (non-zero vorticity) initial perturbations of a constant state, even arbitrarily small ones, it is completely unknown whether global smooth solutions exist or whether singularities form in finite time; the best known result is the lifespan lower bound T≥c/δT\ge c/\delta when the initial vorticity has size δ\delta (Hou–Yin 2022). The problem asks to settle this question in either direction.

Background

The isentropic compressible Euler equations in three space dimensions govern the density ρ(t,x)>0\rho(t,x)>0 and velocity v(t,x)∈R3v(t,x)\in\mathbb{R}^3 of an inviscid fluid: ∂tρ+div(ρv)=0\partial_t\rho+\mathrm{div}(\rho v)=0 and ∂t(ρv)+div(ρv⊗v)+∇p(ρ)=0\partial_t(\rho v)+\mathrm{div}(\rho v\otimes v)+\nabla p(\rho)=0, where the pressure p(ρ)p(\rho) is prescribed by an equation of state. For a polytropic gas, p(ρ)=Aργp(\rho)=A\rho^\gamma with γ>1\gamma>1; for a Chaplygin gas, p(ρ)=P0−B/ρp(\rho)=P_0-B/\rho with constants P0,B>0P_0,B>0 (a pressure law used in cosmological dark-energy models and in the theory of zero-pressure flows). The vorticity is curl v\mathrm{curl}\,v; data with curl v0≡0\mathrm{curl}\,v_0\equiv 0 are called irrotational, in which case the velocity stays a gradient v=∇φv=\nabla\varphi and the system can be reduced, via Bernoulli's law, to a single quasilinear wave equation for the potential φ\varphi.

A characteristic field of a hyperbolic system is called genuinely nonlinear if its eigenvalue varies along the corresponding characteristic direction (which compresses characteristics and produces shocks) and linearly degenerate otherwise; the system is totally linearly degenerate (TLD) if every characteristic field is linearly degenerate. The 3D Euler system of a Chaplygin gas is the standard prototype of a multidimensional TLD symmetric hyperbolic system. In one space dimension, TLD systems are known to have global smooth solutions for small data. On page 89 of his 1984 book, A. Majda posed the conjecture that in dimensions d≥2d\ge 2 a TLD nonlinear symmetric system 'typically' has smooth global solutions for initial data in Hs(Rd)H^s(\mathbb{R}^d) with s>d/2+1s>d/2+1, so that shock formation never occurs; as Majda emphasized, resolving this would isolate how genuine nonlinearity is responsible for shock formation in quasilinear hyperbolic systems. The conjecture remains open even for small data.

For the 3D Chaplygin system the known positive cases all exploit irrotationality or symmetry. For small, compactly supported, irrotational data the potential equation satisfies the null condition (a cancellation structure making the quadratic interactions weak at null infinity), and global smooth existence follows from the classical small-data theory for null-form quasilinear waves (Christodoulou 1986); Gao–Yin (2026) recently extended the irrotational global theory to weakly decaying (non-compactly-supported) data. In two dimensions, Hou–Yin (2019) proved global smooth existence for axisymmetric data with non-zero vorticity, and Ding–Xin–Yin (2022) proved it for irrotational short-pulse data (smooth data concentrated in a thin annular region, of the kind introduced in Christodoulou's shock-formation theory), resolving Majda's conjecture 'in this special case'. In the opposite direction, linear degeneracy is known not to preclude every kind of breakdown: in one space dimension the Chaplygin gas admits delta-like singularities formed by envelopes of different characteristic families (Kong–Wei–Zhang 2013), though the small-data 1D theory is global.

For genuinely non-irrotational data in 3D, nothing is known in either direction. The breakthrough of Hou–Yin (2022) established, through weighted energies, pointwise decay estimates, and a good-unknown substitution (replacing the raw velocity unknown by a combination of velocity and sound speed that enjoys faster space-time decay), that for an initial perturbation of size ε\varepsilon of a constant state with initial vorticity of size δ\delta (0<δ≤ε0<\delta\le\varepsilon), the lifespan of the smooth solution satisfies Tδ≥c/δT_\delta\ge c/\delta for the Chaplygin gas (and Tδ≥cmin⁡{e1/ε,1/δ}T_\delta\ge c\min\{e^{1/\varepsilon},1/\delta\} for polytropic gases, where shocks do eventually form). Their paper states explicitly that for the small-perturbation problem of the 3D Chaplygin Euler equations with non-zero vorticity 'it is still completely unknown whether the global smooth solution exists or not', and this remains the situation as of August 2026 (Gao–Yin 2026, Remark 1.4: for d≥2d\ge 2 with non-vanishing initial rotation the conjecture 'is still completely open' except for small symmetric solutions). The obstruction is structural: without irrotationality there is no potential-flow reduction and no null condition available, while at the same time total linear degeneracy rules out the standard shock-formation mechanism, so neither a global-existence proof nor a breakdown mechanism is currently available.

Problem Statement

For the three-dimensional isentropic compressible Euler equations

∂tρ+div(ρv)=0,∂t(ρv)+div(ρv⊗v)+∇p(ρ)=0\partial_t\rho+\mathrm{div}(\rho v)=0,\qquad \partial_t(\rho v)+\mathrm{div}(\rho v\otimes v)+\nabla p(\rho)=0

with the Chaplygin-gas equation of state p(ρ)=P0−B/ρp(\rho)=P_0-B/\rho (P0,B>0P_0,B>0), consider smooth initial data (ρ,v)(0,x)=(ρˉ+ερ~0(x), εv~0(x))(\rho,v)(0,x)=(\bar\rho+\varepsilon\tilde\rho_0(x),\,\varepsilon\tilde v_0(x)) perturbing a constant rest state ρˉ>0\bar\rho>0, v≡0v\equiv 0, where ε>0\varepsilon>0 is small, the perturbation decays at spatial infinity — e.g. is compactly supported, or lies in the weighted Sobolev classes of the source paper (Hou–Yin 2022), where the perturbation (ρ0,u0):=(ερ~0,εv~0)(\rho^0,u^0):=(\varepsilon\tilde\rho_0,\varepsilon\tilde v_0) and the initial vorticity are measured by the norms ε=∑k≤N∥(⟨∣x∣⟩∇)k(ρ0,u0)∥L2\varepsilon=\sum_{k\le N}\|(\langle|x|\rangle\nabla)^k(\rho^0,u^0)\|_{L^2} and δ=∑k≤N−1∥⟨∣x∣⟩(⟨∣x∣⟩∇)k curl u0∥L2\delta=\sum_{k\le N-1}\|\langle|x|\rangle(\langle|x|\rangle\nabla)^k\,\mathrm{curl}\,u^0\|_{L^2} with N≥8N\ge 8 and ⟨∣x∣⟩=1+∣x∣2\langle|x|\rangle=\sqrt{1+|x|^2} — and curl v~0≢0\mathrm{curl}\,\tilde v_0\not\equiv 0, i.e. the data are not irrotational. Determine whether, for all sufficiently small ε\varepsilon, the solution remains smooth for all time — (ρ−ρˉ,v)∈C([0,∞);Hs(R3))∩C1([0,∞);Hs−1(R3))(\rho-\bar\rho,v)\in C([0,\infty);H^s(\mathbb{R}^3))\cap C^1([0,\infty);H^{s-1}(\mathbb{R}^3)) with s>5/2s>5/2 — or whether there exist arbitrarily small smooth non-irrotational data for which a singularity (e.g. blowup of first derivatives, shock formation) develops in finite time. This is the prototype open case of Majda's conjecture for totally linearly degenerate multidimensional symmetric systems; the irrotational case in 3D and the axisymmetric and short-pulse cases in 2D are already settled positively, and the known lower bound on the lifespan is Tδ≥c/δT_\delta\ge c/\delta when the initial vorticity has size δ\delta, so a resolution in either direction also determines whether, for the data class considered here, that lower bound can be improved to global existence.

References

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