1 Introduction↩︎

The formation of finite-time singularities for the two-dimensional inviscid Boussinesq equations and for the three-dimensional incompressible Euler equations with swirl remains among the central open problems in mathematical fluid dynamics. In this paper we study closed \((1+2)\)-dimensional subsystems \((E1,E2)\), rigorously derived from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations in velocity–pressure form under a parity ansatz on the meridian plane. Thus we discover a unification because the integer \(m=1,2\) only appears in two numerical coefficients of the unified system \((Em)\). Our second emphasis in this revision is on two exact unified \((1+1)\)D systems, denoted \((R0)\) and \((Z0)\), which are rigorously derived from \((Em)\) by restricting to the distinguished symmetry axes. Our aim is twofold: first, to identify a precise apex finite-time blow-up mechanism already visible in these exact \((1+1)\)D descendants of the (2D Boussinesq, 3D axisymmetric Euler) systems; second, to formulate a perturbative stability theory around compatible backgrounds that is mathematically solid at the linear level and explicit about the remaining nonlinear obstruction.

A central advantage of the pressure–velocity formulation is that the divergence-free condition remains visible throughout the reduction and the symmetry-axis geometry is revealed directly. In the rev5 setup, the evenness in \((r,z)\) propagates and therefore preserves the ridge-flatness condition automatically on the axes \(\theta=0,\pm \frac{\pi}{2}\). Although the convective terms do not vanish identically on those axes away from the origin, the apex dynamics at \(x=0\) is closed and decoupled from the off-apex region. This yields two exact unified \((1+1)\)D systems \((R0)\) and \((Z0)\) from the full Euler/Boussinesq-derived subsystem \((Em)\). In our view, this exact derivation of \((R0)\) and \((Z0)\) is one of the main rigorous outputs of the paper, and it deserves to be emphasized at least as strongly as the later conditional stability framework. The resulting analysis is therefore best described as a pressure–velocity approach to finite-time blow-up for two unified \((1+1)\)D systems rigorously derived from the 2D inviscid Boussinesq and the 3D axisymmetric Euler equations.

We call the Hou–Li type variables \(\{u,v,g\}\) of 2 the building blocks of vorticity, because their physical dimensions agree with those of vorticity. In these variables, the quadratic stretching terms also simplify to \((uv,\, v^2-u^2,\, -g^2)\), which makes the CLM-type reaction structure transparent.

1.0.0.1 Related work and context.

The mathematical literature around singularity formation for inviscid fluids is extensive, and we recall only the works most directly connected with the reduction, blow-up mechanism, and perturbative framework used here. Classical continuation and loss-of-regularity perspectives for the 3D Euler equations include Beale–Kato–Majda [1] and Constantin [2], [3]. For the inviscid 2D Boussinesq system, local theory and conditional breakdown criteria go back to Cannon–DiBenedetto [4], Chae–Nam [5], Chae–Kim–Nam [6], and Taniuchi [7]; see also Wu’s lecture notes [8] and the small-scale formation work of Kiselev–Park–Yao [9]. For the axisymmetric Euler geometry, the pressure–velocity and vorticity-building-block viewpoint is naturally compared with the standard Euler and axisymmetric formulations in Majda–Bertozzi [10], Hou–Li [11], Chae–Lee [12], Drazin–Riley [13], and Chen–Fang–Zhang [14].

Model problems and exactly solvable mechanisms provide an important guide for the present apex dynamics. The Constantin–Lax–Majda model [15], the De Gregorio model [16], Schochet’s viscous model analysis [17], the didactic 2D model of Chae–Constantin–Wu [18], the Boussinesq-type one-dimensional model of Choi–Kiselev–Yao [19], the axisymmetric Euler model of Choi–Hou–Kiselev–Luo–Šverák–Yao [20], and the Hou–Liu one-dimensional axisymmetric scenario [21] all illustrate how a reduced stretching law can isolate a finite-time blow-up mechanism. Recent rigorous PDE singularity and perturbative-stability frameworks include Elgindi–Jeong [22], [23], Chen–Hou [24], [25], Drivas–Elgindi [26], and Elgindi–Pasqualotto [27]. The weighted-energy and blow-up-stability language used later is also close in spirit to the broader self-similar and ODE-blow-up stability literature, including Giga–Kohn [28], Giga [29], Merle–Raphaël [30], Raphaël–Rodnianski [31], Collot–Merle–Raphaël [32], and Khenissy–Zaag [33], as well as the local regularity and weighted-estimate tradition represented by Caffarelli–Kohn–Nirenberg [34] and Lin [35].

Compared with these works, the present paper starts from the pressure–velocity form of both the 2D inviscid Boussinesq equations and the 3D axisymmetric Euler equations, works with smooth functions on the full reduced-plane geometry, and uses parity and symmetry-axis structure rather than boundary effects or lower-regularity singular norms to expose the reduced dynamics. This viewpoint keeps the divergence-free constraint visible throughout the derivation and leads to the unified system \((Em)\), in which the integer \(m=1,2\) enters only through two numerical coefficients. The exact axis/apex restrictions then yield the two unified \((1+1)\)D systems \((R0)\) and \((Z0)\); these are not ad hoc model equations, but closed descendants of the Euler/Boussinesq-derived subsystem. The conditional background–remainder theory later in the paper should therefore be read as a perturbative transfer mechanism for this rigorously derived apex blow-up structure.

Main achievements.

• We derive the closed subsystems \((E1,E2)\) exactly from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations under a parity ansatz and identify the variables \(\{u,v,g\}\) as convenient vorticity building blocks.

• We rigorously derive from \((Em)\) two exact unified \((1+1)\)D systems, \((R0)\) and \((Z0)\), carried by the symmetry axes. These systems are not model approximations but exact axis restrictions of the 2D inviscid Boussinesq and 3D axisymmetric Euler reduction.

• We show that the pressure–velocity form reveals the divergence-free structure and the symmetry-axis geometry in a way that is compatible with the exact ray reduction.

• We derive the exact remainder equations around a prescribed background in the \((x,\theta)\) variables, with all pure-background contributions retained in the background system.

• We prove weighted singular linear estimates and formulate a conditional nonlinear remainder theorem of Elgindi type: once a compatible background, the weighted elliptic input, and a subcritical gap exponent \(\sigma\in(C_{\rm lin},1)\) are available, blow-up transfers from its apex dynamics to the full solution.

• We isolate the remaining open step in the program, namely the construction and control of a full background away from the apex together with the compatibility structure needed to close the nonlinear bootstrap.

Organization. Section 2 derives the polar \((1+2)\)D subsystems \((E1,E2)\) from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations and identifies the exact symmetry-axis reductions. Section 3 studies the closed apex ODE system CLM-\(q\) and proves its finite-time blow-up profile. Section 4 treats the convective axis reduction \((R0)\) and shows that the same explicit apex blow-up persists at \(x=0\). Section 5 derives the exact background–remainder system around a prescribed background. Section 6 records the background coefficient bounds and late-time scales needed for the perturbative argument. Section 7 specifies the initial and boundary conditions imposed on the remainder variables. Section 8 proves the weighted energy inequalities and formulates the conditional blow-up transfer mechanism. Section 9 summarizes the rigorous results already obtained and isolates the remaining gap to a full 2D inviscid Boussinesq and 3D axisymmetric Euler blow-up theorem. Section 10 records acknowledgements and provenance remarks.

2 The derivation of system (Em) from 2D inviscid Boussinesq equations and the 3D axisymmetric Euler equations↩︎

2.1 Velocity–pressure formulation and Hou–Li type variables↩︎

In this section we simultaneously convert the velocity–pressure form of the 2D inviscid Boussinesq equations (see, for example, Wu [8], Elgindi–Jeong [23], and Kiselev–Park–Yao [9]) and the 3D axisymmetric Euler equations (see, for example, Chae–Lee [12], Drazin–Riley [13], Chen–Fang–Zhang [14]) into a new formulation in terms of vorticity building blocks. In this form, the structure of the vortex-stretching and convection terms becomes transparent, which makes the study of the apex blow-up mechanism in compressed coordinates more tractable.

In the velocity-pressure form, the 2D inviscid Boussinesq equations for velocity \(\boldsymbol{u}=u_2\boldsymbol{e}_2+u_3\boldsymbol{e}_3\), pressure \(P\) and buoyancy scalar \(\vartheta\) in \((x_2,x_3)\in\mathbb{R}^2\) and the 3D axisymmetric Euler equations on the semimeridian plane \((r\geq 0, z\in\mathbb{R})\) are given by

It is well-known (see e.g. Majda and Bertozzi [10]) that 2D Boussinesq equations 1 have properties similar to the 3D axisymmetric Euler equations 2, at least away from the symmetry axis (\(r=0\)). Indeed, comparing 1 with 2, we see that the buoyancy force \(\vartheta\) in the Boussinesq equation plays the role of the axisymmetric Euler centrifugal force \((rv^\phi)^2/r^3\). Equivalently, after division by the axis coordinate, the Boussinesq quantity \(u^2=\vartheta/x_2\) corresponds to the Euler quantity \((v^\phi/r)^2\). The real difference between the two systems only emerges near the axis of symmetry, where the factors of \(r\) can conceivably change the nature of the dynamics.

In this section, we make the analogy precise by removing the caveat “away from the symmetry axis.”

For 2D inviscid Boussinesq equations, we assume that \(u_2\) is odd in \(x_2\) and even in \(x_3\), that \(u_3\) is even in \(x_2\) and odd in \(x_3\), that \(P\) is even in \((x_2,x_3)\), and that \(\vartheta\) is odd in \(x_2\) and even in \(x_3\). Define the Hou–Li [11] type variables \[\label{eq:uvg-def-B} \{v,g,u^2,p\}:=\left\{-\frac{u_2}{x_2},\frac{u_3}{x_3},\frac{\vartheta}{x_2},P\right\}.\tag{1}\] Then 1 can be converted to the \((m=1)\) version \((E1)\) of system \((Em)\) 3 (with \((r,z)=(x_2,x_3)\) for notational convenience).

For 3D axisymmetric Euler equations, we assume that \(\left(v^{\phi},v^{r}\right)\) are odd in \(r\) and even in \(z\), while \(v^z\) is even in \(r\) and odd in \(z\), and \(P\) is even in \((r,z)\). Define the Hou–Li [11] type variables by \[\label{eq:uvg-def} \{v,u,g,p\}:=\left\{-\frac{v^r}{r},\frac{v^{\phi}}{r},\frac{v^z}{z},P\right\}.\tag{2}\] Then, as shown in Shi [36], 2 can be converted to the \((m=2)\) version \((E2)\) of system \((Em)\) 3 :

\[\label{eq:Em} \left\{ \begin{align} &\tfrac{\mathrm{D}}{\mathrm{D}t}u =\tfrac12m^2uv,\qquad\qquad t\in[0,T), (r,z)\in\mathbb{R}^2\\ &\tfrac{\mathrm{D}}{\mathrm{D}t}v =v^2-u^2+\tfrac{1}{r}p_r\\ &\tfrac{\mathrm{D}}{\mathrm{D}t}g =-g^2\,\,\,\,\,-\tfrac{1}{z}p_z\\ &z\partial_z g-r\partial_r v+g-mv=0,\\ &\tfrac{\mathrm{D}}{\mathrm{D}t}:=\partial_t-v r\partial_r+g z\partial_z. \end{align} \right.\tag{3}\]

2.2 Constantin–Lax–Majda equations↩︎

We now present the explicit finite-time blow-up solutions of the Constantin–Lax–Majda system as a benchmark for further comparison.

Constantin, Lax, and Majda converted 3 into the scalar complex ODE with dependent variable \(z(t,x)=v(t,x)+i\,u(t,x)\) and found the explicit solution:

Substituting the initial data into 4 yields the following result.

If \(u^2\) is solved from 3 (2) and substituted into 3 (1), then one obtains a second order ODE for \(v(t,x)\). Setting \(\tau=6t\), \(v(t,x)=V(\tau,x)\), and \(u(t,x)=U(\tau,x)\), we can convert this ODE to the \(q=2\) version of the following ODE (CLM-\(q\)): \[\label{eq:CLM-q} \left\{ \begin{align} V_{\tau\tau}&=VV_\tau-\tfrac{q}{2(q+1)^2}V^3,\\ V(0)&=V_0,\\ V_\tau(0)&=\tfrac{1}{2(q+1)}\bigl(V_0^2-U_0^2\bigr) \end{align}\right.\tag{4}\]

2.3 System (E\(m\))↩︎

We now introduce a stream function \(\bar \psi\) and augment 3 with two additional relations, obtaining system 5 . To reserve the symbols \((u,v,g,p)\) for later perturbation variables, we place bars on the background unknowns. Thus system (E\(m\)) in 5 consists of five dependent variables \((\bar u,\bar v,\bar g,\bar p,\bar \psi)\), viewed as even functions of \((r,z)\) on the meridian plane, together with seven equations; the last one defines \(\tfrac{\mathrm D}{\mathrm Dt}\). For future convenience, we also introduce \(t\)-scaling parameter \(\lambda\) and \(z^2\)-scaling parameter \(\mu\).

\[\label{E2} \left\{\begin{align} 0&=\tfrac{\mathrm D}{\mathrm Dt}\,\bar u - \tfrac12m^2\,\bar u\,\bar v, \qquad(t,r,z)\in [0,T)\times\mathbb{R}^2\\ 0&=\tfrac{\mathrm D}{\mathrm Dt}\,\bar v -\bar v^{2}+\bar u^{2}-\tfrac{1}{r}\bar p_{r}, \\ 0&=\tfrac{\mathrm D}{\mathrm Dt}\,\bar g +\bar g^{2}+\tfrac{\mu}{z}\,\,\bar p_{z},\\ 0&=z\partial_z\bar g-r\partial_r\bar v+\bar g-m\bar v, \\ 0&=\bar v-\bar\psi-z\partial_z\bar\psi, \\ 0&=\bar g-m\bar\psi-r\partial_r\bar\psi, \\ \tfrac{\mathrm D}{\mathrm Dt}:&=\lambda\partial_t-\bar v\,r\partial_r+\bar g\,z\partial_z. \end{align}\right.\tag{5}\]

For the 3D axisymmetric Euler/Navier-Stokes equations, the components of the vorticity vector \(\boldsymbol{\omega}=\nabla\times \boldsymbol{v}\) are given by

Thus

Equation 6 shows how the building blocks of the vorticity are at work.

2.4 Polar coordinates \((x=\sqrt{r^2+z^2},\;\theta=\arctan(z/r))\)↩︎

We use polar coordinates on the meridian plane: \[\label{eq:polarRxi} r=x \cos\left(\theta\right), \quad z=x \sin\left(\theta\right).\tag{6}\]

2.5 Background↩︎

We write the background solutions as \[\label{eq:background-0} \begin{align} \bar u&=U(t,x,\theta),\quad \bar v=V(t,x,\theta),\quad\bar g=G(t,x,\theta),\quad \bar p=P(t,x,\theta). \end{align}\tag{7}\]

After substituting 27 into the first four equations in 5 , we obtain four equations with the following structure:

\[\label{eq:ZERO-1234} \left\{ \begin{align} \lambda U_t-\tfrac12m^2V\,U&=-x U_{x}W\qquad\quad\,\,\,+J_1\cdot K_1(U_\theta),\\[2mm] \lambda V_{t}-V^2+U^2&=-x V_{x} W+\tfrac{1}{x}P_{x}\,\,\,+J_2\cdot K_2(V_\theta,P_\theta),\\[2mm] \lambda G_{t}+G^2&=-x G_{x} W-\tfrac{\mu}{x}P_{x}\,\,+J_3\cdot K_3(G_\theta,P_\theta),\\[2mm] G-mV&=-x \partial_x W\quad\qquad\,\,\,+J_4\cdot K_4(V_\theta+G_\theta),\\ W:&=G\sin ^2(\theta ) -V\cos ^2(\theta ). \end{align} \right.\tag{8}\]

where \[\label{eq:ZERO-1234-2} \left\{ \begin{align} J_1&=\bigl\{-\tfrac12\sin(2\theta ) (G+V)\bigr\},\\ K_1&=\{U_\theta\},\\[2mm] J_2&=\bigl\{-\tfrac12\sin(2\theta)(G+V),-\tfrac{\tan(\theta)}{x^2}\bigr\}\\ K_2&=\{V_\theta,P_\theta\},\\[2mm] J_3&=\bigl\{-\tfrac12\sin (2\theta )(G+V),-\tfrac{\mu\cot(\theta)}{x^2}\bigr\},\\ K_3&=\{G_\theta,P_\theta\}\\[2mm] J_4&=\bigl\{-\tfrac12\sin (2\theta ),-\tfrac12\sin (2\theta )\bigr\},\\ K_4&=\{V_\theta,G_\theta\}. \end{align} \right.\tag{9}\]

We now examine how the equations simplify under the following ridge-flat Ansatz (in the directions normal to the ridge): \[\label{eq:flat} \left\{\begin{align} &(P_\theta,V_\theta,U_\theta,G_\theta)|_{\theta_0}=0,\qquad\theta_0=0,\pi,\quad x\geq 0,\quad t\in [0,T),\\[2mm] &(P_\theta,V_\theta,U_\theta,G_\theta)|_{\theta_1}=0,\qquad\theta_1=\pm\tfrac{\pi}{2},\quad x\geq 0,\quad t\in [0,T). \end{align}\right.\tag{10}\]

2.6 Symmetry axes and axis-restricted functions↩︎

3 Phase portrait of the apex ODE system↩︎

We consider the nonlinear second–order ODE, CLM-\(q\), ?? . This \(1+1\)D system is not presented as an independent model; rather, it is the exact pointwise ODE satisfied at the apex \(x=0\) on the symmetry axes, with the \(x\to\infty\) limit retained only as a formal asymptotic consistency check. \[\label{eq:ode} v_{tt}=v\,v_t-\alpha v^3, \qquad \alpha=\tfrac{q}{2(1+q)^2},\tag{13}\] with initial data \[\label{eq:ic} v(0)=a\in\mathbb{R}, \qquad v_t(0)=\tfrac{1}{2(1+q)}(a^2-b^2), \quad b^2=\tfrac{2q+m}{m(m+1)}u_0^2\ge 0,\tag{14}\] and throughout this manuscript we assume \[q=m+1>1,\qquad m=1,2.\] Thus for 2D inviscid Boussinesq, we have \(\bigl(m=1,q=2>\tfrac{m^2}{2}\bigr)\). This model reduces to the exact CLM-\(2\) type (cf. 4 ). For 3D axisymmetric Euler, we have \(\bigl(m=2,q=3>\tfrac{m^2}{2}\bigr)\). This model reduces to the exact CLM-\(3\) type.

3.1 Velocity renormalization and first integral↩︎

Define \[\label{eq:wdef} w(t)=\frac{2(1+q)v_t(t)}{v(t)^2}, \qquad v_t=\frac{w}{2(1+q)}v^2.\tag{15}\] On any monotone interval with \(v\neq0\), one can regard \(w\) as a function \(w(v)\).

3.2 Compactification for general \(q>1\)↩︎

The phase variable \(w\) has a coordinate singularity at \(v=0\) because \(w\propto v_t/v^2\). To visualize trajectories through (or toward) \(v=0\), we use the compactified variable \[\label{eq:wbar95def} \bar w:=\frac{w}{1+|w|}\in(-1,1).\tag{16}\] Then: \[w\to+\infty \Longleftrightarrow \bar w\to 1,\qquad w\to-\infty \Longleftrightarrow \bar w\to -1, \qquad w=0 \Longleftrightarrow \bar w=0.\] The distinguished levels \(w=1\) and \(w=q\) map to finite horizontal levels \[\label{eq:wbar95levels} \bar w(1)=\frac{1}{2},\qquad \bar w(q)=\frac{q}{1+q}\in\left(\frac{1}{2},1\right).\tag{17}\] Thus the \((v,\bar w)\)-plane compactifies both the blow-up \(w\to\pm\infty\) and the dynamically important lines \(w=1\), \(w=q\) into a bounded strip.

3.3 General turning amplitude for \(q>1\)↩︎

3.4 A general formula for the first turning time \(t_3\) and its small-\(b\) asymptotics↩︎

In this section we assume \[\label{eq:assume95t3} q>1,\qquad a>0,\qquad 0<b^2<a^2,\tag{18}\] so that \(v_t(0)=\frac{1}{2(1+q)}(a^2-b^2)>0\) and \(w_0=1-b^2/a^2\in(0,1)\). We define \(t_3=t_3(q,a,b)\) to be the first time the trajectory reaches the turning locus \(w=0\) on the \(v>0\) branch, i.e. \[(v,\bar w)=\bigl(|v_{\mathrm{turn}}|,0^+\bigr).\]

3.5 A closed quadrature in the \(w\)-variable↩︎

For any \(q>1\), combining \(dt/dv=\frac{2(1+q)}{w v^2}\) with ?? gives \[\label{eq:dt95dw95general} \frac{dt}{dw} =\frac{dt}{dv}\frac{dv}{dw} = -\,\frac{1+q}{(w-1)(w-q)}\cdot \frac{1}{v(w)}.\tag{19}\] On the \(v>0\), \(w\in(0,1)\) branch, the invariant ?? reads \[\label{eq:v95of95w95general} v(w)^{2(q-1)} = C_*\,\frac{1-w}{(q-w)^q}, \qquad (0<w<1),\tag{20}\] hence \[\label{eq:v95of95w95general95root} v(w) =\left(C_*\right)^{\! \frac{1}{2(q-1)}}\left(\frac{1-w}{(q-w)^q}\right)^{\! \frac{1}{2(q-1)}}.\tag{21}\] Substituting 21 into 19 and using \((w-1)(w-q)=(1-w)(q-w)\) for \(w\in(0,1)\) yields:

3.6 Small-\(b\) limit: \(t_3\to 2(q+1)/a\)↩︎

3.7 The integrable benchmark \(q=2\): explicit verification and the clock picture↩︎

For \(q=2\), \(\alpha=1/9\) and 13 becomes \[v_{tt}=v v_t-\frac{1}{9} v^3, \qquad v_t(0)=\frac{1}{6}(a^2-b^2).\] Set \(A=a^2+b^2\). The exact solution is \[\label{eq:q295v} v(t)=-\frac{6\bigl(b^2t+a(at-6)\bigr)}{(at-6)^2+b^2t^2} =-\frac{6(At-6a)}{A t^2-12at+36}.\tag{22}\] The phase variable and compactification are \[\label{eq:q295w95wbar} w(t)=1-\frac{36b^2}{(At-6a)^2}, \qquad \bar w(t)=\frac{w(t)}{1+|w(t)|}.\tag{23}\] The turning amplitude is \(|v_{\mathrm{turn}}|=\dfrac{A}{2| b|}\).

Subcase (A1): \(a>0\), \(a^2>b^2\) — 3–6–9–12 clockwise↩︎

Assume \(a>0\) and \(a^2>b^2\) (so \(a>|b|\)). Define \[t_3=\frac{6(a-|b|)}{A},\qquad t_6=\frac{6a}{A},\qquad t_9=\frac{6(a+|b|)}{A}.\] Then \((v(t),\bar w(t))\) hits \[\bigl(|v_{\mathrm{turn}}|,0^+\bigr)\;\text{at }t=t_3,\qquad (0^+,-1)\;\text{at }t=t_6,\qquad \bigl(-|v_{\mathrm{turn}}|,0^-\bigr)\;\text{at }t=t_9,\] with the timeline \(0<t_3<t_6<t_9<\infty\). As \(t\to\infty\), \((v(t),\bar w(t))\to(0^-,1/2)\), the “12 o’clock” mark, and the last leg (9 to 12) takes infinite time.

Subcase (B1): \(a<0\), \(a^2>b^2\) — a one-sided arc↩︎

Assume \(a<0\) and \(a^2>b^2\). Then \(v(t)<0\) for all \(t\ge0\), \(v(t)\uparrow 0^-\) as \(t\to\infty\), and \(\bar w(t)\uparrow 1/2\). In the clock picture this corresponds to a single clockwise arc from about “10 o’clock” toward “12 o’clock”.

3.8 Proof of Theorem 14↩︎

We give a self-contained proof of the finite-time blow-up characterization in Theorem 14, using the phase-portrait machinery developed above.

3.9 Pointwise reduction in \(x\) and the two cases \(b(x)=0\) vs.\(b(x)\ne 0\)↩︎

Fix \(x\) and abbreviate \[a:=a(x)=V(0,x,\theta_*),\qquad b:=b(x)=U(0,x,\theta_*),\] where \(\theta_*\) denotes one of the symmetry-axis angles. Along the corresponding ridge, the \((U,V)\)-subsystem reduces to the CLM-\(q\) ODE (equivalently 13 –14 after eliminating \(u\)), so the question of blow-up is pointwise in \(x\).

Case 1: \(b=0\). When \(b=0\), the ridge equation forces \(U(t,x,\theta_*)\equiv 0\) by uniqueness, and the \(v\)-equation reduces to the Riccati ODE \[\label{eq:riccati95b0} v_t=\frac{1}{2(q+1)}\,v^2,\qquad v(0)=a.\tag{24}\] Hence \[\label{eq:riccati95solution} v(t)=\frac{a}{1-\frac{a}{2(q+1)}t}.\tag{25}\] If \(a>0\), then \(v(t)\to+\infty\) at the finite time \(T(x)=\frac{2(q+1)}{a}\); moreover the \(U\)-component stays identically zero along the ridge. If \(a\le 0\), then the denominator in 25 never vanishes for \(t\ge 0\) and \(v(t)\) remains bounded (indeed \(v(t)\uparrow 0\) if \(a<0\) and \(v\equiv 0\) if \(a=0\)). Thus, at a fixed \(x\), finite-time blow-up occurs if and only if \(b(x)=0\) and \(a(x)>0\).

Case 2: \(b\ne0\). Assume \(b^2>0\) and \(q>1\). If \(a=0\), then \(v_t(0)<0\); for every sufficiently small \(\tau>0\), one has \(v(\tau)<0\) and \(v(\tau)\ne0\), so the first-integral argument below applies after restarting the ODE at \(t=\tau\). We may therefore assume \(a\ne0\). Then \(w_0=1-b^2/a^2<1\) (see ?? ), so the trajectory on the \((v,w)\)-plane starts in the strip \(0<w<1\) when \(a^2>b^2\), or in \(w<0\) when \(a^2<b^2\). Lemma 1 provides the first integral ?? , and Lemma 2 shows that any turning point satisfies \(|v|\le |v_{\mathrm{turn}}|<\infty\). In particular, on the \(v>0\) branch with \(a>0\) and \(a^2>b^2\), the solution reaches the turning locus \(w=0\) in finite time \(t_3=t_3(q,a,b)\) (Theorem 16), at which point \(v(t_3)=v_{\mathrm{turn}}\) is finite. After \(t_3\), the vector field in 13 drives the orbit through the remaining “clockwise” quadrants in the compactified phase plane, but the invariant ?? prevents \(w\) from escaping to \(+\infty\) while \(v\) stays bounded by \(v_{\mathrm{turn}}\). Consequently, \(v\) remains bounded for all \(t\ge 0\), and by the algebraic relation between \(u\) and \((v,v_t)\) (obtained by solving the second equation of the CLM-\(q\) system for \(u^2\)), the \(u\)-component is bounded as well. Standard ODE continuation therefore yields a global classical solution.

3.10 Earliest blow-up over \(x\)↩︎

Define the set \[Z:=\{x:\;b(x)=0\;\text{and}\;a(x)>0\}.\] By the pointwise analysis above, blow-up occurs at some \(x\) if and only if \(Z\neq\emptyset\). For each \(x\in Z\), the blow-up time is \(T(x)=\frac{2(q+1)}{a(x)}\), hence the earliest blow-up time is obtained by maximizing \(a\) over \(Z\): \[T=\inf_{x\in Z}T(x)=\frac{2(q+1)}{\max_{x\in Z}a(x)}.\] Let \(\bar x\in Z\) attain the maximum (as in Theorem 14). Then \(V(t,\bar x)\) blows up at \(t=T\) and no other \(x\) can blow up earlier. This completes the proof of Theorem 14.

4 A convective axis reduction: the system \((R0)\)↩︎

In this section we study the convective axis reduction \((R0)\) ?? associated with the symmetry-axis dynamics of the Euler-reduced system. Unlike the closed apex ODE ?? , convection remains present away from the origin; nevertheless, all convective terms vanish at \(x=0\), so the exact apex blow-up mechanism persists there. We impose the initial data \[\label{eq:R0-data} u(0,x)=B x^2 e^{-B_1x^2}, \qquad v(0,x)=A e^{-A_1x^2}, \qquad A,B,A_1,B_1>0.\tag{26}\]

The key observation is that, for even classical solutions, the symmetry center \(x=0\) is dynamically closed: all the explicitly \(x\)-weighted convective terms vanish there, and the trace at \(x=0\) satisfies exactly the same positive ODE system analyzed in Section 3. Thus the apex blow-up mechanism survives even though the full axis-restricted dynamics is no longer convection free away from the origin.

5 Perturbation PDEs↩︎

We derive the remainder system around a prescribed background whose ridge/apex behavior is the one identified above. Throughout this section we work in the first-quadrant polar variables \[x\geq0,\qquad \theta\in\left[0,\tfrac{\pi}{2}\right],\] with \[r=x\cos\theta,\qquad z=x\sin\theta.\] This choice is consistent with the rev5 geometry: the axes of the first quadrant are the boundary lines \(\theta=0,\tfrac{\pi}{2}\). The full solutions are written as the sum of the background fields and the remainders: \[\label{eq:full95def} \left\{\begin{align} &\bar u=U(t,x,\theta)+u(t,x,\theta),\\ &\bar v=V(t,x,\theta)+v(t,x,\theta),\\ &\bar g=G(t,x,\theta)+g(t,x,\theta),\\ &\bar p=P(t,x,\theta)+p(t,x,\theta),\\ \end{align}\right.\tag{27}\]

Setting \(q=m+1\) in ?? , we obtain \[\label{eq:lambda-mu-2} \left\{\begin{align} &\lambda=\lambda(m)=\tfrac{m+2}{m+1}m^2,\qquad \mu=\mu(m)=\tfrac{3m+2}{2(m+1)-m^2}m^2,\\ &m=1,\quad q=2,\quad\lambda=\tfrac{3}{2},\quad \mu=\tfrac{5}{3},\\ &m=2,\quad q=3,\quad\lambda=\tfrac{16}{3},\quad \mu=16.\\ \end{align}\right.\tag{28}\]

In order to simplify the notation, we still use the symbols \((\lambda,\mu)\) in the rest of the paper and hide the explicit \(m\)-dependence as shown in 28 .

After substituting 27 into 5 (1,2,3,4), we separate the full system into the background equations for \((V,U,G,P)\) and the exact remainder equations for \((v,u,g,p)\). All pure-background terms are kept in the background equations, so the remainder system contains only linear couplings to the background and genuinely nonlinear remainder–remainder interactions:

\[\label{eq:zero-1234} \left\{\begin{align} \lambda u_{t}&=\tfrac{m^2}{2}(u V+vU)-\tfrac12\bigl(U_{\theta} (g+v)+u_{\theta} (G+V)\bigr)\sin (2\theta ) \\ &-\bigl(x U_{x} g+x u_{x} G\bigr)\sin ^2(\theta )+\bigl(x U_{x} v+x u_{x} V\bigr)\cos ^2(\theta ) +N_1\\[2mm] \lambda v_{t}&=2 (vV-uU)+\tfrac{1}{x}p_{x}-\tfrac{\tan (\theta ) }{x^2}p_{\theta }-\tfrac12\bigl(v_{\theta} (G+V)+V_{\theta }(g+v)\bigr)\sin (2\theta ) \\ &-\bigl(x v_{x} G+x V_{x}g\bigr)\sin ^2(\theta )+\bigl(x V_{x} v+x v_{x} V\bigr)\cos ^2(\theta ) +N_2\\[2mm] \lambda g_{t}&=-2 gG-\tfrac{\mu}{x}p_{x}-\tfrac{\mu\cot (\theta ) }{x^2}p_{\theta }-\tfrac12\bigl(g_{\theta} (G+V)+G_{\theta }(g+v)\bigr)\sin (2\theta ) \\ &-\bigl(x g_{x} G+x G_{x}g\bigr)\sin ^2(\theta )+\bigl(x G_{x} v+x g_{x} V\bigr)\cos ^2(\theta )+N_3,\\[2mm] mv-g&=x g_{x}\sin ^2(\theta )-x v_{x}\cos ^2(\theta ) +\tfrac12\sin (2\theta ) \bigl(g_{\theta }+v_{\theta }\bigr). \end{align}\right.\tag{29}\]

where the nonlinear perturbation terms are given by \[\label{eq:nonlinear-123} \left\{\begin{align} N_1:&=\tfrac{m^2}{2}uv\quad-\tfrac12\sin(2\theta)(g+v)u_\theta-xu_x\bigl(g\sin^2(\theta)-v\cos^2(\theta)\bigr),\\[2mm] N_2:&=v^2-u^2-\tfrac12\sin(2\theta)(g+v)v_\theta-xv_x\bigl(g\sin^2(\theta)-v\cos^2(\theta)\bigr),\\[2mm] N_3:&=-g^2\quad\,\,\,-\tfrac12\sin(2\theta)(g+v)g_\theta-xg_x\bigl(g\sin^2(\theta)-v\cos^2(\theta)\bigr). \end{align}\right.\tag{30}\]

The divergence-free condition for the perturbation \((v,g)\) 29 (4) can be solved with the stream function \(\psi(t,x,\theta)\) defined below: \[\label{eq:vg-psi} \left\{\begin{align} v&=\,\,\,\psi+x\psi_x\sin^2(\theta)+\tfrac12\psi_\theta\sin(2\theta),\\ g&=m\psi+x\psi_x\cos^2(\theta)-\tfrac12\psi_\theta\sin(2\theta). \end{align}\right.\tag{31}\]

These two equations are the polar coordinate version of 5 (5,6).

5.1 Getting rid of \((p_\theta,p_x)\)↩︎

Our next step is to get rid of \((p_\theta,p_x)\). Define \(\omega\) as \[\label{eq:Omega-omega-def} \begin{align} \omega:&=\tfrac12\sin(2\theta)(xg_x+\mu xv_x)-g_\theta\sin^2\theta +\mu v_\theta\cos^2\theta,\\ \end{align}\tag{32}\] For \(0<\theta<\pi/2\), this gives the interior reconstruction formula \[\label{eq:gx-Vx} \begin{align} g_x&=\tfrac{2}{x\sin(2\theta)}\Bigl(\omega-\tfrac{1}{2}\sin(2\theta)\mu xv_x+g_\theta\sin^2\theta -\mu v_\theta\cos^2\theta\Bigr).\\ \end{align}\tag{33}\] The final equations below are written in terms of \(\omega\) and \(\psi\); their displayed coefficients involve only \(\sin\theta\) and \(\cos\theta\), with the angular endpoints handled by the boundary conditions and weighted elliptic estimate.

Now we rewrite 29 \((2,3)\) as: \[\label{eq:px-ptheta} \left\{\begin{align} \lambda v_t&=A+N_2+\tfrac{1}{x} p_x-\tfrac{\tan\theta}{x^2}p_\theta,\\ \lambda g_t&=B+N_3-\tfrac{\mu}{x}p_x-\tfrac{\mu\cot\theta}{x^2}p_\theta,\\ \end{align}\right.\tag{34}\]

where \((A,B)\) are defined as \[\label{eq:AB-def} \left\{\begin{align} A:&=2 (vV-uU)-\tfrac12\bigl(v_{\theta} (G+V)+V_{\theta }(g+v)\bigr)\sin (2\theta ) \\ &-\bigl(x v_{x} G+x V_{x}g\bigr)\sin ^2(\theta )+\bigl(x V_{x} v+x v_{x} V\bigr)\cos ^2(\theta ),\\[2mm] B:&=-2 gG-\tfrac12\bigl(g_{\theta} (G+V)+G_{\theta }(g+v)\bigr)\sin (2\theta ) \\ &-\bigl(x g_{x} G+x G_{x}g\bigr)\sin ^2(\theta )+\bigl(x G_{x} v+x g_{x} V\bigr)\cos ^2(\theta ). \end{align}\right.\tag{35}\]

The solutions for \((p_x,p_\theta)\) then become \[\label{eq:px-ptheta-2} \left\{ \begin{align} \tfrac{1}{x}p_x&=-\cos ^2(\theta ) \left(A+N_2-\lambda v_{t}\right)+\tfrac{1}{\mu}\sin ^2(\theta ) \left(B+N_3-\lambda g_{t}\right)\\[2mm] \tfrac{1}{x^2}p_\theta&=\tfrac{1}{2\mu}\sin (2\theta ) \bigl(\mu A+B+\mu N_2+N_3-\lambda g_{t}-\mu \lambda v_{t}\bigr) \end{align} \right.\tag{36}\]

Using \(p_{x\theta}=p_{\theta x}\) to get rid of \(p\), we obtain:

and using the definition of \(\omega\) in 32 to simplify the result, we finally obtain:

where

Substituting \((A,B)\) of 35 into 10 (1), using the identities obtained from 33 by differentiation to simplify the results, and using 31 to express \((v,g)\) in terms of \((\psi,\psi_x,\psi_\theta)\), we obtain

Substituting \(N_2\), \(N_3\) of 30 into 10 (2), using the identities obtained from 33 by differentiation to simplify the results, and using 31 to express \((v,g)\) in terms of \((\psi,\psi_x,\psi_\theta)\), we obtain \[\label{eq:M2} \left\{ \begin{align} M_2&=-2 \mu u \left(x u_{x}\sin\theta+u_{\theta }\cos (\theta ) \right)\\ &\quad-\tfrac12\sin (2 \theta ) \omega _{\theta }\bigl((m+1) \psi +x \psi _{x}\bigr)\\ &\quad+x \omega _{x} \Bigl(\left(\cos ^2(\theta )-m \sin ^2(\theta )\right) \psi +\tfrac12\sin (2\theta ) \psi _{\theta}\Bigr)\\ &\quad-(m-1) \omega \Bigl(\sin^2 (\theta )x\psi _{x}+\tfrac12\sin (2\theta ) \psi _{\theta}+\psi \Bigr). \end{align}\right.\tag{37}\]

Using 31 to express \((v,g)\) in terms of \((\psi,\psi_x,\psi_\theta)\), the equation for \(u_t\) in 29 (1) can also be converted into the desired form: \[\label{eq:ut} \begin{align} \lambda u_t&= L_1+M_1,\\ \end{align}\tag{38}\] \[\label{eq:L1} \left\{ \begin{align} 2L_1&=m^2 u V+2 x u_{x} \left(\cos ^2(\theta ) V-\sin ^2(\theta ) G\right)-\sin (2 \theta ) u_{\theta }(G+V)\\ &\quad+\psi \left(m^2 U-(m+1) \sin (2 \theta ) U_{\theta }+2 x \left(\cos ^2(\theta )-m \sin ^2(\theta )\right) U_{x}\right)\\ &\quad+x \psi _{x} \left(m^2 \sin ^2(\theta ) U-\sin (2 \theta ) U_{\theta }\right)+\tfrac{1}{2} \sin (2 \theta ) \psi _{\theta } \left(m^2 U+2 x U_{x}\right). \end{align} \right.\tag{39}\] \[\label{eq:M1} \left\{ \begin{align} M_1 &=\tfrac{m^2}{4}u \left(\psi _{\theta}\sin(2\theta) +2 x \psi _{x}\sin ^2(\theta ) +2 \psi \right)\\ &\quad+\tfrac{1}{2} x u_{x} \left(\psi _{\theta}\sin(2\theta) +\psi((m+1) \cos (2 \theta )-m+1) \right)\\ &\quad-\tfrac{1}{2} u_{\theta}\sin(2\theta) \left(x \psi _{x}+(m+1) \psi \right). \end{align} \right.\tag{40}\]

Substitution of 31 into 32 leads to \[\label{eq:omega-tilde-2} \left\{ \begin{align} \omega:&=\Delta \psi,\\ \Delta:&=c_1(\theta)x^2 \partial_{xx}+c_2(\theta) x \partial_x+c_3(\theta)x\partial_{x\theta}+c_4(\theta)\partial_\theta+c_5(\theta)\partial_{\theta\theta}.\\ \end{align}\right.\tag{41}\] where \[\label{eq:Delta-coeff} \left\{ \begin{align} c_1&=\tfrac12\sin (2\theta )\left(\mu \sin ^2(\theta )+\cos ^2(\theta )\right),\\ c_2&=\tfrac{1}{4} \sin (2\theta )((\mu-1) \cos (2 \theta )+5 \mu+2 m+3),\\ c_3&=(\mu-1) \sin (\theta ) \sin (2 \theta ) \cos (\theta ),\\ c_4&=2 \mu \cos ^4(\theta )+\sin ^2(\theta ) (\cos (2 \theta )-m),\\ c_5&=\tfrac12\sin (2\theta ) \left(\mu \cos ^2(\theta )+\sin ^2(\theta )\right). \end{align}\right.\tag{42}\]

6 Initial energy bounds↩︎

6.1 Initial energy and finiteness↩︎

We record the “initial energy” (at \(t=0\)) associated with the background profile: \[\label{eq:initial-energy} \begin{align} E(0):&=\int_{0}^{\pi/2}\int_{0}^{\infty}(x\cos\theta)^2\Bigl(U(0,x,\theta)^2+V(0,x,\theta)^2\Bigr)\,x^2\,dx\,d\theta\\ &\quad+\int_{0}^{\pi/2}\int_{0}^{\infty}(x\sin\theta)^2G(0,x,\theta)^2\,x^2\,dx\,d\theta. \end{align}\tag{43}\]

Using the initial conditions similar to those in ?? : \[\label{initial-condition-UVG-2} \left\{\begin{align} U(0,x,\theta)&=Bx^2\exp\bigl(-B_1 x^2(1+B_2\phi(\theta))\bigr),\quad B,B_1,B_2>0\\ V(0,x,\theta)&=A\exp\bigl(-A_1 x^2(1+A_2\phi(\theta))\bigr),\quad A,A_1,A_2>0\\ G(0,x,\theta)&=mV(0,x,\theta). \end{align}\right.\tag{44}\]

We immediately deduce that the initial energy is bounded on the first-quadrant wedge \(x\ge0\), \(\theta\in[0,\tfrac{\pi}{2}]\).

7 Elliptic problem and initial/boundary conditions for the perturbation PDEs↩︎

7.1 Updated Linear PDEs and Nonlinear Terms↩︎

The final remainder system for \((u,\omega,\psi)\) is:

\[\label{eq:linear} \left\{\begin{align} \lambda u_t&=L_1+M_1\quad\quad \eqref{eq:L1},\eqref{eq:M1}\\ \lambda\omega_{t}&={L}_{2}+{M}_2\quad\quad\eqref{eq:L2},\eqref{eq:M2},\\ \omega&=\Delta \psi.\qquad\qquad\eqref{eq:omega-tilde-2}\\ \end{align}\right.\tag{45}\]

7.2 Coefficient Functions↩︎

The effective elliptic operator \(\Delta\) in 41 (2) is defined by: \[\label{eq:omega-tilde-2-b} \Delta=c_1(\theta)x^2 \partial_{xx}+c_2(\theta) x \partial_x+c_3(\theta)x\partial_{x\theta}+c_4(\theta)\partial_\theta+c_5(\theta)\partial_{\theta\theta}.\tag{46}\] where \[\label{eq:Delta-coeff-3} \left\{ \begin{align} c_1&=\tfrac12\sin (2\theta )\left(\mu \sin ^2(\theta )+\cos ^2(\theta )\right),\\ c_2&=\tfrac{1}{4} \sin (2\theta )((\mu-1) \cos (2 \theta )+5 \mu+2 m+3),\\ c_3&=\tfrac12(\mu-1) \sin ^2(2 \theta ),\\ c_4&=2 \mu \cos ^4(\theta )+\sin ^2(\theta ) (\cos (2 \theta )-m),\\ c_5&=\tfrac12\sin (2\theta ) \left(\mu \cos ^2(\theta )+\sin ^2(\theta )\right). \end{align}\right.\tag{47}\] Notice that \[\label{eq:omega-tilde-2-c} \left\{\begin{align} &\omega(t,x,0)=\bigl(\Delta \psi\bigr)_{\theta=0}=2\mu\psi_{\theta}(t,x,0),\\ &\omega(t,x,\tfrac{\pi}{2})=\bigl(\Delta \psi\bigr)_{\theta=\pi/2}=-(m+1)\psi_{\theta}(t,x,\tfrac{\pi}{2}).\\ \end{align}\right.\tag{48}\]

7.3 Initial conditions and boundary conditions↩︎

We select the first quadrant \((x\geq 0, \theta\in [0,\tfrac{\pi}{2}])\) as our domain. The boundary conditions (for the angular edges and for \(x\to\infty\)) for the perturbations \((u,v,g)\) are:

\[\label{eq:boundary-condition-0} \left\{\begin{align} &u(t,x,\theta)\bigr|_{\theta=0,\pi/2}=0,\quad u(t,x,\theta)\to0 \text{ as } x\to\infty,\\ &v(t,x,\theta)\bigr|_{\theta=0,\pi/2}=0,\quad v(t,x,\theta)\to0 \text{ as } x\to\infty,\\ &g(t,x,\theta)\bigr|_{\theta=0,\pi/2}=0,\quad g(t,x,\theta)\to0 \text{ as } x\to\infty, \end{align}\right.\tag{49}\] Thus the background dynamics on the angular edges at \(\theta=0,\tfrac{\pi}{2}\) are not disturbed.

Using 31 , we have \[\label{eq:vg-psi-2} \left\{\begin{align} 0=v(t,x,0)&=\Bigl(\psi+x\psi_x\sin^2(\theta)+\tfrac12\psi_\theta\sin(2\theta)\Bigr)_{\theta=0}\\ &=\psi(t,x,0),\\ 0=g(t,x,0)&=\Bigl(m\psi+x\psi_x\cos^2(\theta)-\tfrac12\psi_\theta\sin(2\theta)\Bigr)_{\theta=0}\\ &=m\psi(t,x,0)+x\psi_x(t,x,0),\\ 0=v(t,x,\tfrac{\pi}{2})&=\Bigl(\psi+x\psi_x\sin^2(\theta)+\tfrac12\psi_\theta\sin(2\theta)\Bigr)_{\theta=\pi/2}\\ &=\psi(t,x,\tfrac{\pi}{2})+x\psi_x(t,x,\tfrac{\pi}{2}),\\ 0=g(t,x,\tfrac{\pi}{2})&=\Bigl(m\psi+x\psi_x\cos^2(\theta)-\tfrac12\psi_\theta\sin(2\theta)\Bigr)_{\theta=\pi/2}\\ &=m\psi(t,x,\tfrac{\pi}{2}),\\ \end{align}\right.\tag{50}\]

The combination of 49 and 50 leads to the boundary conditions \[\label{eq:boundary-condition} \left\{\begin{align} &u(t,x,\theta)\bigr|_{\theta=0,\pi/2}=0,\quad u(t,x,\theta)\to0 \text{ as } x\to\infty,\\ &\psi(t,x,\theta)\bigr|_{\theta=0,\pi/2}=0,\quad \psi(t,x,\theta)\to0 \text{ as } x\to\infty,\\ &\psi_x(t,x,\theta)\bigr|_{\theta=0,\pi/2}=0,\quad \psi_x(t,x,\theta)\to0 \text{ as } x\to\infty, \end{align}\right.\tag{51}\]

The initial conditions are given by \[\label{eq:init-condition} \left\{\begin{align} &u(0,x,\theta)\text{ admits a smooth even extension in }x\text{ across }x=0,\\ &\psi(0,x,\theta)\text{ admits a smooth even extension in }x\text{ across }x=0,\\ &u(0,x,\theta),\;\psi(0,x,\theta) \text{ vanish sufficiently fast as } \theta\to 0,\tfrac{\pi}{2}. \end{align}\right.\tag{52}\]

The phrase “sufficiently fast vanishing” as \(\bigl(\theta\to0,\tfrac{\pi}{2}\bigr)\) means enough vanishing and regularity near the edge so that the weighted Sobolev norms used below are finite and the boundary terms produced by integration by parts vanish at \(\bigl(\theta=0,\tfrac{\pi}{2}\bigr)\).

8 Linear estimates and conditional nonlinear control up to blow-up time↩︎

Updated perturbation system. Throughout this section we work with the final perturbation equations derived in Section 5. We study the perturbation system 38 and 9, together with the coefficient collections 39 ,40 ,11,37 and the elliptic operator 46 , on the time interval \([0,T)\) up to the background blow-up ridge apex time, around the prescribed background ?? . Throughout, all Lebesgue and Sobolev norms are taken with respect to the weighted measure \(d\mu_w=w(\theta)x^2\,dx\,d\theta\), where \(w(\theta)\) is a fixed positive angular weight chosen to encode the endpoint behavior. The stability norms use the adapted derivatives \[Z_x:=x\partial_x,\qquad D_\theta := \partial_\theta.\] For a function \(f\), the notation \(H^s_{\mu_w,Z}\) denotes the weighted Sobolev norm generated by the derivatives \(Z_x^jD_\theta^\ell f\) with \(j+\ell\le s\).

8.1 Bootstrap framework and adapted background coefficient bounds↩︎

Fix an integer \(k\ge 6\). Define the perturbation energy \[\label{eq:stab-energy} \mathcal{E}_k(t):= \sum_{\substack{j+\ell\le k}}\Big(\|Z_x^j D_\theta^\ell u(t)\|_{L^2_{\mu_w}}^2+\|Z_x^j D_\theta^\ell \omega(t)\|_{L^2_{\mu_w}}^2\Big) +\sum_{\substack{j+\ell\le k+1}}\|Z_x^j D_\theta^\ell \psi(t)\|_{L^2_{\mu_w}}^2.\tag{53}\]

Background coefficient bounds actually needed in the energy method. The ridge-background construction based on the seed 44 provides the closed-form/apex model for \((U,V)\) used here and, in particular, reproduces the explicit apex dynamics. What the stability estimates require is not a uniform bound on the raw derivatives \(\partial_x^j\partial_\theta^\ell V\) (which can grow faster than \((T-t)^{-1}\) near the intermediate scale \(r^2\sim T-t\)), but rather uniform control of the degenerate combinations that appear in 39 ,11 and in the adapted weighted Sobolev norms.

In particular, \[\|V(t)\|_{L^\infty}\lesssim \frac{1}{T-t},\qquad \|U(t)\|_{L^\infty}\lesssim \frac{1}{T-t}.\]

8.2 Elliptic control of \(\psi\) from \(\omega=\Delta\psi\)↩︎

The elliptic relation in 45 reads \(\omega=\Delta\psi\), where \(\Delta\) is given by 46 . After rewriting the angular part in terms of the adapted derivative \(D_\theta\) (as already indicated in the weighted-norms subsection), the operator \(\Delta\) is treated as a weighted elliptic operator in the variables \((\log x,\theta)\), with endpoint weights carried by the coefficients in 47 . Accordingly, we record the following weighted elliptic estimate as the analytic input needed for the perturbation argument: for all integers \(s\ge 0\), \[\label{eq:elliptic-new} \|\psi(t)\|_{H^{s+2}_{\mu_w,Z}}\le C_{\Delta,s}\,\|\omega(t)\|_{H^{s}_{\mu_w,Z}},\tag{54}\] where the constant depends only on the wedge geometry, the boundary conditions, and the weighted coefficient structure appearing in 46 . This estimate is natural from the \(y=\log x\) reformulation discussed in Remark 23; in the present manuscript we use it as a working elliptic input for the \(\psi\)-estimate rather than as a separately proved theorem.

In particular, since \(k\ge 6\), Sobolev embedding in the \((x,\theta)\) variables (with \(D_\theta\) counted as one derivative) gives \[\label{eq:linf-from-E} \|u(t)\|_{L^\infty}+\|\omega(t)\|_{L^\infty}+\|\psi(t)\|_{W^{1,\infty}_Z} \le C\,\mathcal{E}_k(t)^{1/2}.\tag{55}\]

8.3 Energy inequality for \((u,\omega)\)↩︎

Differentiate the \(u\)-equation and the \(\omega\)-equation in 45 by \(Z_x^j D_\theta^\ell\) for \(j+\ell\le k\), take the \(L^2_{\mu_w}\) inner product with \(Z_x^j D_\theta^\ell u\) and \(Z_x^j D_\theta^\ell \omega\), and sum over \(j+\ell\le k\). The transport terms are now written directly in the \((x,\theta)\) variables, so the integration-by-parts step is carried out in \(x\) and \(\theta\). The boundary contributions vanish because of the remainder boundary conditions at \(\bigl(\theta=0,\tfrac{\pi}{2}\bigr)\), the decay as \(x\to\infty\), and the weighted formulation using \(D_\theta=\partial_\theta\).

Using the commutator estimates and 55 , one obtains an inequality of the form \[\label{eq:Ek-ineq} \frac{d}{dt}\mathcal{E}_k(t) \le \frac{C_{\rm lin}}{T-t}\,\mathcal{E}_k(t) + C_{\rm nl}\,\Big(\|M_1(t)\|_{H^k_{\mu_w,Z}}+\|M_2(t)\|_{H^k_{\mu_w,Z}}\Big)\,\mathcal{E}_k(t)^{1/2}.\tag{56}\]

Quadratic remainder terms. From the explicit forms of \(M_1,M_2\) in 40 and 37 , together with Moser and Sobolev product estimates in the \((x,\theta)\) variables, one obtains \[\|M_1(t)\|_{H^k_{\mu_w,Z}}+\|M_2(t)\|_{H^k_{\mu_w,Z}} \le C\,\mathcal{E}_k(t).\] Because all pure-background terms have been kept in the background system, there is no additive forcing term in the remainder energy inequality. Thus it is natural to rewrite 56 in terms of \[Y(t):=\mathcal{E}_k(t)^{1/2}.\] Then \[\label{eq:Y-ineq-E2} Y'(t)\le \frac{C_{\rm lin}}{T-t}Y(t)+C_{\rm nl}Y(t)^2\tag{57}\] whenever \(Y(t)>0\).

The important point is that 57 by itself does not yet imply a closed bootstrap with a remainder strictly smaller than the background singularity. What it does give is an Elgindi-type conditional transfer principle: if the remainder stays in a class whose growth is weaker than the background blow-up rate, then the quadratic term is perturbative and the background singularity transfers to the full solution.

To make this precise, fix an exponent \(\sigma>0\) and define the renormalized energy envelope \[X_\sigma(t):=(T-t)^\sigma Y(t).\] Differentiating and using 57 gives \[\label{eq:Xsigma-ineq} X_\sigma'(t) \le \frac{C_{\rm lin}-\sigma}{T-t}X_\sigma(t)+C_{\rm nl}(T-t)^{-\sigma}X_\sigma(t)^2.\tag{58}\] Hence, whenever \[\label{eq:sigma-gap} \sigma>C_{\rm lin},\tag{59}\] and whenever a bootstrap bound of the form \[\label{eq:Xsigma-bootstrap} X_\sigma(t)\le M\varepsilon \qquad\text{for }0\le t\le t_*\tag{60}\] holds with \(\varepsilon>0\) sufficiently small, the right-hand side of 58 is integrable and the quadratic term can be absorbed. Standard continuity then yields \[\label{eq:Xsigma-close} X_\sigma(t)\le 2X_\sigma(0) \qquad\text{for }0\le t\le t_*.\tag{61}\] Equivalently, \[\label{eq:Y-sigma-bound} Y(t)\le 2Y(0)\Bigl(\frac{T}{T-t}\Bigr)^\sigma, \qquad 0\le t\le t_*.\tag{62}\] In particular, if one can choose \(\sigma<1\) while still having 59 , then the remainder stays strictly below the background blow-up scale \((T-t)^{-1}\) in the detecting norm.

This discussion is summarized in the following conditional theorem.

Under this conditional control, one obtains a blow-up transfer statement for the full solution.

Accordingly, the logical bottleneck of the manuscript is no longer a forcing obstruction in the remainder equations. The main unresolved issue is instead the rigorous construction/control of a background away from the apex, with the coefficient bounds needed by the weighted energy method and with enough rigidity near the apex to match the explicit apex dynamics, together with whatever refined estimate is needed to produce a genuine gap exponent \(\sigma<1\) in the remainder norm. Once those two inputs are available, the present stability mechanism upgrades directly to a nonlinear remainder theorem in the spirit of Elgindi.

9 Conclusion↩︎

We derived closed \((1+2)\)D subsystems \((E1,E2)\) from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations and showed that \((Em)\) contains two exact unified \((1+1)\)D descendants, \((R0)\) and \((Z0)\), obtained by restricting to the distinguished symmetry axes. This exact derivation is one of the main rigorous achievements of the manuscript. The rev5 geometry preserves ridge flatness automatically through the evenness in \((r,z)\), and at the apex \(x=0\) the dynamics closes exactly. Thus the finite-time singularity mechanism is already visible at the level of these rigorously derived unified \((1+1)\)D systems before one turns to the full conditional background–remainder analysis.

The weighted energy method developed in Section 8 shows that, if a compatible background exists on \([0,T)\) with the coefficient bounds required there and with apex trace governed by the closed apex dynamics, and if the remainder stays subordinate to the background singularity in the detecting norm, then the full solution inherits the same finite-time blow-up.

The main unresolved step is therefore the construction and control of a full background away from the apex, together with the rigidity properties needed to match the apex dynamics and close the nonlinear bootstrap without loss. The blow-up mechanism itself is explicit at the ridge/apex level, but extending that information to a full background with the necessary compatibility bounds remains the decisive open problem.

Even before the final nonlinear theorem is completed, the present formulation already isolates the core components of the analysis. It provides an exact derivation from 3D axisymmetric Euler, a precise ridge/apex blow-up mechanism, a strong linearized stability framework, conditional nonlinear control, and a conditional blow-up transfer statement.

Natural next steps are therefore clear. The first is to prove the full background existence/control theorem compatible with the apex dynamics identified here. The second is to sharpen the detecting norm so that the remainder remains strictly below the background blow-up rate, yielding a closed nonlinear bootstrap. After that, one can revisit modulation of geometric parameters and lower-regularity weighted theories.

10 Acknowledgements↩︎

ChatGPT is credited as a substantive contributor to drafting and technical editing; responsibility for correctness remains with the author. The author thanks Prof. Zixiang Zhou of the Department of Mathematics at Fudan University and Prof. Jie Qin of the Department of Mathematics at the University of California, Santa Cruz for their continuous support and encouragement over the years. The author also thanks colleagues and the broader PDE and fluid-dynamics community for stimulating discussions on axisymmetric Euler and CLM-type models. (Computational assistance: OpenAI’s GPT-5.5 Pro through ChatGPT; sessions in March–April 2026.)

References↩︎