2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)
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| Format: | Preprint |
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2026
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| _version_ | 1866911639375183872 |
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| author | Shi, Yaoming |
| author_facet | Shi, Yaoming |
| contents | We derive $(1+2)$D subsystems~$(E1,E2)$ from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations in the (meridian) plane. The integer $m=1,2$ only appears in two numerical coefficients of subsystem~$(Em)$. Thus we discover a unification. We then study two unified $(1+1)$-dimensional systems, denoted $(R0)$ and $(Z0)$, that are rigorously derived from the $(Em)$. The main point of view in this revision is that these $(1+1)$D systems are not ad hoc model equations and not merely ``symmetry-axis reductions.'' Rather, they arise as exact symmetry-axis/apex restrictions of the full $(1+2)$D system~$(Em)$ obtained from 2D inviscid Boussinesq and 3D axisymmetric Euler, and they already contain the core finite-time singularity mechanism of the full problem.
The paper has three main outputs. First, it derives the polar $(1+2)$D subsystem~$(Em)$ from the 2D inviscid Boussinesq equations and from the 3D axisymmetric Euler equations and identifies the exact unified $(1+1)$D systems $(R0)$ and $(Z0)$ carried by the symmetry axes. Second, it proves finite-time blow-up for the resulting apex dynamics and analyzes the associated convective axis reduction. Third, it derives the exact background--remainder equations and formulates a conditional nonlinear stability mechanism: if a compatible full background exists on $[0,T)$ with the adapted coefficient bounds required by the weighted energy method, if the weighted elliptic estimate holds, and if a gap exponent $σ\in(C_{\rm lin},1)$ is available so that the remainder remains below the background scale, then the same finite-time apex blow-up transfers to the full solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26715 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$) Shi, Yaoming Analysis of PDEs Exactly Solvable and Integrable Systems Fluid Dynamics Primary 35B44, 35Q86, 76B03, Secondary 35B35, 35Q35 We derive $(1+2)$D subsystems~$(E1,E2)$ from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations in the (meridian) plane. The integer $m=1,2$ only appears in two numerical coefficients of subsystem~$(Em)$. Thus we discover a unification. We then study two unified $(1+1)$-dimensional systems, denoted $(R0)$ and $(Z0)$, that are rigorously derived from the $(Em)$. The main point of view in this revision is that these $(1+1)$D systems are not ad hoc model equations and not merely ``symmetry-axis reductions.'' Rather, they arise as exact symmetry-axis/apex restrictions of the full $(1+2)$D system~$(Em)$ obtained from 2D inviscid Boussinesq and 3D axisymmetric Euler, and they already contain the core finite-time singularity mechanism of the full problem. The paper has three main outputs. First, it derives the polar $(1+2)$D subsystem~$(Em)$ from the 2D inviscid Boussinesq equations and from the 3D axisymmetric Euler equations and identifies the exact unified $(1+1)$D systems $(R0)$ and $(Z0)$ carried by the symmetry axes. Second, it proves finite-time blow-up for the resulting apex dynamics and analyzes the associated convective axis reduction. Third, it derives the exact background--remainder equations and formulates a conditional nonlinear stability mechanism: if a compatible full background exists on $[0,T)$ with the adapted coefficient bounds required by the weighted energy method, if the weighted elliptic estimate holds, and if a gap exponent $σ\in(C_{\rm lin},1)$ is available so that the remainder remains below the background scale, then the same finite-time apex blow-up transfers to the full solution. |
| title | 2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$) |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems Fluid Dynamics Primary 35B44, 35Q86, 76B03, Secondary 35B35, 35Q35 |
| url | https://arxiv.org/abs/2603.26715 |