Global solvability with nonzero initial vorticity

Prove the global solvability of the three-dimensional compressible isentropic Euler equations for Chaplygin gases with smooth, compactly supported, small initial perturbations having nonvanishing initial vorticity, thereby resolving the stated conjecture beyond the irrotational case.

Background

The paper's analysis relies on the irrotationality condition rot⁡v0≡0\operatorname{rot}v_0\equiv0, which permits reduction of the Euler system to a scalar quasilinear wave equation for a velocity potential. This leaves the genuinely rotational case outside the scope of the proved global existence theorem.

Remark \ref{Rem-3} states that, for compactly supported smooth data with nonzero initial vorticity in dimensions d≥2d\ge2, the general conjecture is completely open apart from certain small symmetric solutions. The three-dimensional nonzero-vorticity case is therefore an explicitly unresolved problem.

References

When $(\rho_0(x), v_0(x))\in C_0{\infty}(\Bbb Rd)$ with $d\ge 2$ and $\rot v_0(x)\not\equiv 0$, so far the {\bf Conjecture} is still completely open except the global solvability on the small symmetric solution of Euler with pressure2-irrot:condition (see - and ).

Euler:

$\left\{ \begin{aligned} &_t\rho+\Div(\rho v)=0,\\ &_t(\rho v)+\Div(\rho v \otimes v)+\nabla p=0,\\ \end{aligned} \right. $

pressure2:

p(ρ)=P0−Dρ,p(\rho)=P_0-\frac{D}{\rho},

irrot:condition:

(ρ,v)(0,x)=(ρˉ+ρ0(x),v0(x)),\begin{split} &(\rho,v)(0,x)=(\bar\rho+\rho_0(x), v_0(x)),\\ \end{split}

— Global classical solutions to 3D irrotational compressible Euler equations of Chaplygin gases with weakly decaying initial data  (2608.13166 - Gao et al., 13 Aug 2026) in Remark 1.3 (labeled Remark \ref{Rem-3}) in Section 1, subsection "Main results and remarks"