By the center manifold theorem (Carr-Jones) [12] [13] [14] [6], there exists a 3D invariant manifold,

$$ \mathcal{M}^c(0) = \{ \vec{Y} \in \mathbb{R}^5 \mid \vec{Y}(\xi) \to \vec{Y}_0 \text{ as } \xi \to -\infty \}, \qquad (483) $$

tangent to the center eigenspace at $\vec{Y}_0$.

## 11.8. Stable Modes and End-State Analysis

For $r \to \infty$, the system reduces to

$$ \frac{d\vec{Y}}{dr} = A(r)\vec{Y} + \vec{g}(r), \quad \|\vec{g}(r)\| \ll \|A(r)\vec{Y}\|. \qquad (484) $$

Let $\text{spec}(A(r)) \to \{\lambda_i^\infty\}_{i=1}^5$ as $r \to \infty$. Assume $\Re(\lambda_i^\infty) < 0$ for $i = 1, \dots, d$, then the flow admits a $d$-dimensional stable manifold,

$$ \mathcal{M}^s(\infty) = \{ \vec{Y} \in \mathbb{R}^5 \mid \vec{Y}(r) \to 0 \text{ as } r \to \infty \}, \quad \dim \mathcal{M}^s = d. \qquad (485) $$

## 11.9. Global Matching via Invariant Manifolds

A global profile corresponds to a heteroclinic connection,

$$ \vec{Y}(r) \in \mathcal{M}^c(0) \cap \mathcal{M}^s(\infty). \qquad (486) $$

If the intersection is transverse, the solution set forms a smooth manifold of dimension

$$ \dim(\mathcal{M}^c(0) \cap \mathcal{M}^s(\infty)) = \dim \mathcal{M}^c + \dim \mathcal{M}^s - 5. \qquad (487) $$

Codimension-zero intersections yield open families of admissible blow-up profiles. Tangencies induce bifurcations, marking qualitative transitions in solution structure.

## 12. Stability via Lyapunov Functionals

The original compressible MHD system admits the classical energy functional (10). In the inviscid, ideal case ($\mu = \nu = 0$), this functional is formally conserved, reflecting the total mechanical and magnetic energy. In the presence of dissipation, $\mathcal{E}[t]$ satisfies

$$ \frac{d}{dt} \mathcal{E}[t] \le 0, \qquad (488) $$

which motivates the introduction of a Lyapunov-type functional in similarity variables.

**Theorem 12.1 (Monotonicity of a Lyapunov functional).** Let $(\vec{U}, \vec{H}, R)$ be a sufficiently regular solution to the self-similar MHD system derived from the ansatz (87), and let $\mathcal{E}_s[t]$ denote the similarity-scaled energy,

$$ \mathcal{E}_s[t] = \int_{\mathbb{R}^3} \left( \frac{1}{2} R |\vec{U}|^2 + \frac{|\vec{H}|^2}{2\mu_0} + \mathcal{P}(R \| R_\infty) \right) d\vec{X}, \qquad (489) $$

where $\mathcal{P}(R \| R_\infty)$ is defined as in (92).

Then $\mathcal{E}_s[t]$ is a strict Lyapunov functional: it is non-increasing in time, and strictly decreasing unless $(\vec{U}, \vec{H}) \stackrel{\text{eqv}}{=} 0$ and $R = R_\infty$.

So we have the differential identity,

$$ \frac{d}{dt} \mathcal{E}_s[t] = -\mu \int |\nabla \vec{U}|^2 d\vec{X} - \nu \int |\nabla \vec{H}|^2 d\vec{X} \le 0. \qquad (490) $$

*Proof.* Differentiating under the integral sign and using the self-similar system (271) (272) (276), we arrive at

$$ \frac{d}{dt} \mathcal{E}_s[t] = \int \left( R \vec{U} \cdot \partial_t \vec{U} + \frac{1}{\mu_0} \vec{H} \cdot \partial_t \vec{H} + \frac{\delta \mathcal{P}}{\delta R} \partial_t R \right) d\vec{X}. \qquad (491) $$
Rewriting $\partial_t \vec{U}$, $\partial_t \vec{H}$ and $\partial_t R$ using the self-similar evolution equations and integrating by parts, the convective and pressure terms cancel due to incompressibility constraints and symmetries. The dissipative terms survive,

$$ \frac{d}{dt} \mathcal{E}_s[t] = -\mu \int |\nabla \vec{U}|^2 - \nu \int |\nabla \vec{H}|^2 \le 0. \quad (492) $$

Moreover, equality holds if and only if both velocity and magnetic gradients vanish, which implies $\vec{U}$, $\vec{H}$ constant. By decay and integrability, this forces $(\vec{U}, \vec{H}) = 0$. Then, the mass constraint and pressure condition imply $R = R_\infty$. $\square$

The decay of the Lyapunov functional suggests that the self-similar state may act as an attractor for nearby configurations. To test this, we introduce infinitesimal perturbations and derive the corresponding linearized evolution. Let

$$ R(r, \tau) = R(r) + \epsilon\eta(r, \tau), \quad (493) $$

$$ U(r, \tau) = U(r) + \epsilon v(r, \tau), \quad (494) $$

$$ B(r, \tau) = B(r) + \epsilon w(r, \tau), \quad (495) $$

with $\epsilon \ll 1$ and $\tau = -\log(-t)$ the similarity time. We collect the perturbation fields into a vector,

$$ \vec{\psi}(r, \tau) = \begin{pmatrix} \eta(r, \tau) \\ v(r, \tau) \\ w(r, \tau) \end{pmatrix}, \quad (496) $$

which encapsulates density, velocity, and magnetic perturbations, respectively. Linearizing, we extract the evolution equation

$$ \partial_{\tau} \vec{\psi}(r, \tau) = \lambda \vec{\psi}(r, \tau), \quad (497) $$

where $\lambda$ is a linear (generally non-selfadjoint) differential operator. Its coefficients depend explicitly on the base state $(R(r), U(r), B(r))$ and involve the background non-linearity, radial geometry, and dissipative terms.

## 12.1. Anatomy of the Linear Operator

The operator $\lambda$ arises from the Fréchet derivative of the self-similar MHD equations with respect to the perturbations

$$ \lambda \vec{\psi} = \frac{d}{d\epsilon} \left( \text{RHS}_{\text{self-similar}} \left| R + \epsilon\eta, U + \epsilon v, B + \epsilon w \right| \right) \Big|_{\epsilon=0}, \quad (498) $$

where RHS refers to the right-hand side of the self-similar reduction of the compressible MHD equations, viz. the system of ordinary differential equations governing the radial profiles $R(r)$, $U(r)$, $B(r)$, obtained after inserting the Leray-type ansatz into the full PDE system and simplifying via symmetry (cf. Section 10.2).

The operator $\lambda = \{\lambda_{RR}, \lambda_{RU}, \dots, \lambda_{BB}\}$ acts on the corresponding component of $\vec{\psi}$ through its block matrix introduced in (326).

The diagonal entry $\lambda_{UU}$ contains contributions such as

$$ \lambda_{UU} = -\alpha + U\partial_r - \beta r \partial_r + \mu \left( \partial_r^2 + \frac{2}{r} \partial_r - \frac{2}{r^2} \right), \quad (499) $$

representing advection by the self-similar velocity field, radial scaling, and the viscous Laplacian in spherical symmetry.

## 12.2. Spectral Properties and Gaps

Let $\sigma(\lambda)$ denote the spectrum. The solution is linearly stable if

$$ \sup\{\Re(\lambda) \mid \lambda \in \sigma(\lambda)\} < 0. \quad (500) $$

If there exists $\lambda \in \sigma(\lambda)$ with $\Re(\lambda) > 0$, the mode grows exponentially. Spectral instability is linked to bifurcation and non-uniqueness.
In weighted spaces $L_{\omega}^2$, one can use the Gearhart–Prüss theorem [27] [68] to connect spectral bounds with semigroup decay,

$$ \|e^{\tau \lambda} \|_{B(L_{\omega}^2)} \le C e^{\omega_0 \tau}, \quad \omega_0 = \sup \Re(\sigma(\lambda)), \qquad (501) $$

where the constant $C > 0$ reflects uniform boundedness of the semigroup on bounded time intervals, independently of $\tau$, and depends on resolvent estimates near the spectral bound. The space $\mathfrak{R}(L_{\omega}^2)$ symbolizes the Banach space of bounded linear operators on the weighted Hilbert space $L_{\omega}^2$, equipped with the induced operator norm,

$$ \|T\|_{\mathfrak{R}(L_{\omega}^2)} = \sup_{\|\varphi\|=1} \|T\varphi\|_{L_{\omega}^2}. \qquad (502) $$

## 12.3. Lyapunov Control of Non-linear Behavior

Let

$$ \mathcal{E}_p[\vec{\psi}] = \int_0^\infty (a_1(r)\eta^2 + a_2(r)v^2 + a_3(r)w^2)r^2 dr \qquad (503) $$

be a Lyapunov-type perturbation energy functional, where $\vec{\psi} = (\eta, v, w)$ is the perturbation, with weights $a_i(r)$ determined by the base profiles and dissipative terms. Its time derivative satisfies

$$ \frac{d}{d\tau} \mathcal{E}_p = \langle \vec{\psi}, \lambda \vec{\psi} \rangle + \langle \lambda \vec{\psi}, \vec{\psi} \rangle, \qquad (504) $$

which, in general, is not sign-definite unless $\lambda$ is self-adjoint. This motivates the following heuristic interpretation,

(1) if $\frac{d\mathcal{E}_p}{d\tau} < 0$, then the perturbation decays in similarity time $\tau$, and the profile is linearly stable,

(2) if $\mathcal{E}_p[\vec{\psi}]$ increases over time, the system may be unstable.

Nonetheless, suitable bounds on $\mathcal{E}_p$ may lead to spectral gap estimates, or help rule out unstable modes in restricted function spaces. This approach provides a Lyapunov-type criterion for (in)stability and can complement direct spectral computations.

## 12.4. Essential and Pseudo-Spectral Analysis

In practical settings, one must account for the presence of continuous or essential spectrum due to the unbounded spatial domain $r \in [0, \infty)$. This complicates the spectral picture and may necessitate:

(1) introduction of exponentially weighted function spaces,

(2) numerical discretization of $\lambda$ using spectral or finite-difference methods,

(3) application of pseudo-spectral theory to assess sensitivity of the spectrum.

These tools can be used to determine whether unstable eigenvalues emerge from the essential spectrum or isolated, robust instabilities.

## 12.5. Sobolev Space with Weights

To make these arguments rigorous, we present weighted Lebesgue spaces

$$ L_{\omega}^{2} = \left\{ \varphi : [0, \infty) \rightarrow \mathbb{R} \mid \int_{0}^{\infty} |\varphi(r)|^{2} \omega(r) dr < \infty \right\}, \qquad (505) $$

where the weight is given by

$$ \omega(r) = r^2(1 + r^2)^{\mu}, \quad \mu > 0, \qquad (506) $$

with $\mu$ controlling the integrability rate at infinity. This choice ensures decay of perturbations as $r \to \infty$ and integrability of higher-order terms under radial symmetry.

We designate the space of perturbations as the weighted Sobolev space

$$ \mathfrak{S} = \left\{ \vec{\psi} \in (L_{\omega}^2)^3 \mid \partial_r \vec{\psi} \in (L_{\omega}^2)^3 \right\}, \qquad (507) $$
which is the natural energy space for the radial system. It has the inner product

$$ \langle \vec{\psi}_1, \vec{\psi}_2 \rangle_{\mathfrak{S}} = \sum_{i=1}^{3} \int_{0}^{\infty} [\psi_{1i} \psi_{2i} + \partial_r \psi_{1i} \partial_r \psi_{2i}] \omega(r) dr. \quad (508) $$

This space corresponds to the radial component of the weighted Sobolev space $\mathfrak{S}_{\omega}^{1}([0, \infty); \mathbb{R}^{3})$.

Under appropriate regularity and decay assumptions, one can derive energy inequalities of the form

$$ \frac{d}{d\tau} \|\vec{\psi}\|_{\mathfrak{S}}^2 \leq -\delta \|\vec{\psi}\|_{\mathfrak{S}}^2 + C \|\vec{\psi}\|_{\mathfrak{S}}^p, \quad p > 1, \quad (509) $$

where $C > 0$ is a constant depending on the background solution. Such inequalities imply conditional stability and exponential decay, provided the initial perturbation $\|\vec{\psi}(0)\|_{\mathfrak{S}}$ is sufficiently small.

## 12.6. Variational Principles and Lyapunov Selection

The similarity-reduced system admits a variational formulation in terms of an energy-type functional defined over suitable radially symmetric function spaces. Consider the non-linear functional

$$ \mathcal{E}: \mathcal{D} \subset \mathfrak{S} \to \mathbb{R}, \quad \mathcal{E}[R, U, B] = \int_{0}^{\infty} \left( \frac{1}{2} RU^2 + \frac{1}{2\mu_0} B^2 + A \frac{R^{\gamma_p}}{\gamma_p - 1} \right) r^2 dr, \quad (510) $$

where

(i) $\mathfrak{S}$ is the radial weighted Sobolev space of triplets $(R, U, B) \in (L_{\omega}^{2})^{3}$ with $\partial_{r}(R, U, B) \in (L_{\omega}^{2})^{3}$, and $\omega(r) = r^{2}(1 + r^{2})^{\mu}$,

(ii) $\mathcal{D} \subset \mathfrak{S}$ is the subspace of functions with $R(r) > 0$, and such that $\mathcal{E}[R, U, B] < \infty$,

(iii) $\gamma_p > 1$ and $A > 0$ are constants from the pressure law $p = AR^{\gamma_p}$.

This functional combines kinetic, magnetic and internal energy contributions within the self-similar ansatz. The first variation of $\mathcal{E}$ under admissible perturbations engenders Euler-Lagrange equations that coincide formally with the self-similar ODE system (C) (M) (I).

Let $(R, U, B) \in \mathcal{D}$, and consider smooth compactly supported variations

$$ (R_{\epsilon}, U_{\epsilon}, B_{\epsilon}) = (R + \epsilon\phi, U + \epsilon\psi, B + \epsilon\chi), \quad (511) $$

with $(\phi, \psi, \chi) \in \mathcal{E}_{c}^{\infty}((0, \infty); \mathbb{R}^{3})$. Then the first variation reads

$$ \frac{d}{d\epsilon} \mathcal{E}[R_{\epsilon}, U_{\epsilon}, B_{\epsilon}]|_{\epsilon=0} = \int_{0}^{\infty} \left( \frac{1}{2} U^2 \phi + R U \psi + \frac{1}{\mu_0} B \chi + A R^{\gamma_p - 1} \phi \right) r^2 dr. \quad (512) $$

Demanding that this vanish for arbitrary compactly supported variations recovers balance conditions that, upon identification with the reduced system, reveal a formal variational structure. Precisely,

(1) the continuity equation (C) emerges from variation in $U$,

(2) the momentum equation (M) corresponds to variation in $R$,

(3) the induction equation (I) arises via variation in $B$.

Hence, self-similar profiles may be interpreted as constrained critical points of $\mathcal{E}$, motivating the search for variational minimizers or saddle points. The structure of $\mathcal{E}$ suggests it can serve as a Lyapunov-type functional in the perturbation framework of Section 12.5, potentially providing coercivity and spectral information.

Remark. If $(R, U, B)$ minimizes $\mathcal{E}$ under fixed mass or magnetic helicity constraints, the resulting Euler-Lagrange equations may differ from the unforced self-similar system and correspond to other solution branches or bifurcation points.

## 13. Far-Field Behavior and Matching Theory

We examine the asymptotic structure of self-similar solutions to the compressible MHD equations in the far-field regime $|\vec{X}| \to \infty$. Under finite energy constraints, the decay rates of the fields reflect deep analytic properties of the underlying elliptic system, with implications for both regularity and admissibility.
## 13.1. Calderón–Zygmund-Type Elliptic Estimate

**Lemma 13.1** (Calderón–Zygmund estimate for the pressure gradient). Let $(\vec{U}, \vec{H}, R)$ be a $\mathcal{C}^2$-solution of the stationary similarity system, and postulate $R(\vec{X}) \ge c > 0$ for all $\vec{X} \in \mathbb{R}^3$, with $\vec{U}, \vec{H} \in L^p(\mathbb{R}^3)$ for some $p \in (1, \infty)$. Then the pressure gradient

$$ \nabla \mathcal{P} = -R(\vec{U} \cdot \nabla)\vec{U} + \frac{1}{\mu_0}(\nabla \times \vec{H}) \times \vec{H} \quad (513) $$

obeys the Calderón–Zygmund inequality [10],

$$ \|\nabla \mathcal{P}\|_{L^p(\mathbb{R}^3)} \le C_p \left( \|R\|_{L^\infty} \|\vec{U}\|_{L^{2p}}^2 + \|\vec{H}\|_{L^{2p}}^2 \right), \quad (514) $$

where $C_p > 0$ depends only on $p$ and the elliptic structure.

*Proof.* From Eq. (272)

$$ (\vec{U} \cdot \nabla)\vec{U} + \nabla \left( \frac{P(R)}{R} \right) = \mu \Delta \vec{U} + \frac{1}{\mu_0 R} (\nabla \times \vec{H}) \times \vec{H}, \quad (515) $$

we isolate the pressure gradient,

$$ \nabla \left( \frac{P(R)}{R} \right) = -R(\vec{U} \cdot \nabla)\vec{U} + \frac{1}{\mu_0}(\nabla \times \vec{H}) \times \vec{H} - \frac{\mu}{R} \Delta \vec{U}. \quad (516) $$

Assuming sufficient regularity and $R \ge c > 0$, let $\mathcal{P}(R) = \frac{P(R)}{R} = AR^{\gamma_p-1}$, and compute the divergence,

$$ \Delta \mathcal{P} = \nabla \cdot \left( -R(\vec{U} \cdot \nabla)\vec{U} + \frac{1}{\mu_0}(\nabla \times \vec{H}) \times \vec{H} \right). \quad (517) $$

This is a classical Poisson equation [66] and governs scalar potential fields generated by a source distribution in $\mathbb{R}^3$. The operator $\Delta$ is invertible on tempered distributions modulo constants via the fundamental solution of the Laplacian in $\mathbb{R}^3$, so we write formally,

$$ \mathcal{P}(\vec{X}) = -\frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{1}{|\vec{X} - \vec{Y}|} \nabla_{\vec{Y}} \cdot \left( -R(\vec{Y})(\vec{U} \cdot \nabla)\vec{U} + \frac{1}{\mu_0}(\nabla \times \vec{H}) \times \vec{H} \right) (\vec{Y}) d\vec{Y}. \quad (518) $$

This is a classical singular integral of convolution type with Calderón–Zygmund kernel,

$$ \ker(\vec{X} - \vec{Y}) = \nabla^2 \left( \frac{1}{|\vec{X} - \vec{Y}|} \right), \quad (519) $$

in accordance with the standard size and smoothness conditions.

By Calderón–Zygmund theory on $\mathbb{R}^3$, the operator $\nabla^2 \Delta^{-1}$ is bounded on $L^p$ for $1 < p < \infty$. For that reason,

$$ \|\nabla \mathcal{P}\|_{L^p} \lesssim \left\| R(\vec{U} \cdot \nabla)\vec{U} + (\nabla \times \vec{H}) \times \vec{H} \right\|_{L^p}. \quad (520) $$

The terms on the right are bilinear and bounded under $\vec{U}, \vec{H} \in L^{2p}$ via Hölder-type embeddings,<fnref n="8" /> using the assumed bounds on $R$. This completes the proof. □

## 13.2. Sharp Algebraic Decay under Finite Similarity Energy

**Theorem 13.1** (Asymptotic decay of self-similar profiles). Let $(\vec{U}, \vec{H}, R)$ be a $\mathcal{C}^2$-solution to the stationary similarity system (271) (272) (276) with finite similarity energy,

$$ \int_{\mathbb{R}^3} \left( \frac{1}{2} R |\vec{U}|^2 + \frac{1}{2\mu_0} |\vec{H}|^2 \right) d\vec{X} + \int_{\mathbb{R}^3} \mathcal{P}(R \|R_\infty\|) d\vec{X} < \infty. \quad (521) $$

<footnote n="8">⁸ The continuity exponent traces back to Hölder's study [37] of pointwise regularity via integral means.</footnote>
Then the profiles decay at infinity according to

$$|\vec{U}(\vec{X})| + |\vec{H}(\vec{X})| = \mathcal{O}(|\vec{X}|^{-1}), \qquad (522)$$

$$|R(\vec{X}) - R_{\infty}| = \mathcal{O}(|\vec{X}|^{-1}), \qquad (523)$$

$$\mathcal{P}(R\|R_{\infty})(\vec{X}) = \mathcal{O}(|\vec{X}|^{-2}). \qquad (524)$$

where $\mathcal{O}(|\vec{X}|)$ is the standard big $\mathcal{O}$ behavior at infinity.

*Proof.* Let $\mathfrak{g}_{\rho_{\text{rad}}}(\vec{X})$ be a smooth radial cutoff function such that

$$\mathfrak{g}_{\rho_{\text{rad}}}(\vec{X}) = 1 \text{ for } |\vec{X}| < \rho_{\text{rad}}, \qquad (525)$$

$$\mathfrak{g}_{\rho_{\text{rad}}}(\vec{X}) = 0 \text{ for } |\vec{X}| > 2\rho_{\text{rad}}, \qquad (526)$$

with $|\nabla \mathfrak{g}_{\rho_{\text{rad}}}| \lesssim \rho_{\text{rad}}^{-1}$. Multiply Eq. (272) by $\vec{U} \mathfrak{g}_{\rho_{\text{rad}}}$ and integrate by parts to obtain energy inequalities with controlled boundary error terms.

From the finite energy condition, we deduce that

$$\int_{\mathbb{R}^3} \mathcal{P}(R\|R_{\infty})d\vec{X} < \infty, \qquad (527)$$

and the coercivity estimate

$$\mathcal{P}(R\|R_{\infty})(\vec{X}) \gtrsim |R(\vec{X}) - R_{\infty}|^2 \text{ near equilibrium,} \qquad (528)$$

gives

$$|R(\vec{X}) - R_{\infty}|^2 \lesssim |\vec{X}|^{-2}. \qquad (529)$$

The coercivity of the relative pressure functional,

$$\mathcal{P}(R\|R_{\infty}) = \int_{R_{\infty}}^{R} \frac{P(s) - P(R_{\infty})}{s^2} ds \qquad (530)$$

combined with $P(s) \sim s^{\gamma}$ for $s \gg 1$, provides the asymptotic

$$|R(\vec{X}) - R_{\infty}|^2 \lesssim \mathcal{P}(R\|R_{\infty})(\vec{X}) \lesssim |\vec{X}|^{-2}. \qquad (531)$$

Correspondingly, the decay of $\vec{U}$ and $\vec{H}$ follows from the structure of the similarity Eqq. (272) (276) under the finite energy constraint.

Consider the momentum equation in similarity variables,

$$(\vec{U} \cdot \nabla)\vec{U} - \frac{1}{2}\vec{U} - \frac{1}{2}(\vec{X} \cdot \nabla)\vec{U} + \nabla \left(\frac{P(R)}{R}\right) = \mu \Delta \vec{U} + \frac{1}{\mu_0 R}(\nabla \times \vec{H}) \times \vec{H}. \qquad (532)$$

The non-linear terms decay by energy integrability. Since $R \to R_{\infty}$, the pressure term is determined by

$$\nabla \left( \frac{P(R)}{R} \right) = \mathcal{O}(|\vec{X}|^{-2}), \qquad (533)$$

and the magnetic source complies with

$$\frac{1}{R}(\nabla \times \vec{H}) \times \vec{H} = \mathcal{O}(|\vec{X}|^{-2}). \qquad (534)$$

Thus, we identify a perturbed elliptic equation,

$$\mu \Delta \vec{U} + \text{lower-order terms} = \vec{\phi}(\vec{X}), \quad \vec{\phi} = \mathcal{O}(|\vec{X}|^{-2}). \qquad (535)$$

We now apply a weighted Calderón-Zygmund-type elliptic estimate to conclude

$$|\vec{U}(\vec{X})| \lesssim |\vec{X}|^{-1}, \qquad (536)$$

$$|\vec{H}(\vec{X})| \lesssim |\vec{X}|^{-1}. \qquad (537)$$

To close, recall the continuity equation,

$$\nabla \cdot (R \vec{U}) = 0. \qquad (538)$$
Since $\vec{U} = \mathcal{O}(|\vec{X}|^{-1})$ and $R - R_{\infty} = \mathcal{O}(|\vec{X}|^{-1})$, we can write

$$ \nabla R = \mathcal{O}(|\vec{X}|^{-2}), \qquad (539) $$

which upon integration over rays from infinity gives again $|R(\vec{X}) - R_{\infty}| = \mathcal{O}(|\vec{X}|^{-1})$. $\square$

### 13.3. Asymptotics for $r \to \infty$

To have physical admissibility and finite energy, we impose exponential decay of the form

$$ R(r) \sim r^{-\delta_1} e^{-\lambda_1 r}, \qquad (540) $$

$$ U(r) \sim r^{-\delta_2} e^{-\lambda_2 r}, \qquad (541) $$

$$ B(r) \sim r^{-\delta_3} e^{-\lambda_3 r}, \qquad (542) $$

we obtain leading-order asymptotics consistent with physical expectations and finite energy, with $(\lambda_i, \delta_i)$ determined from the eigenvalues of the Jacobian matrix $J_{\infty} = D\vec{F}(0)$.

These decay rates depend on the linearizing the system at the vacuum fixed point

$$ (R, U, B) = (0, 0, 0), \qquad (543) $$

fulfilling an asymptotic Jacobian matrix

$$ J_{\infty} = D\vec{F}(0). \qquad (544) $$

The decay involves

$$ \lim_{r \to \infty} R(r) = \lim_{r \to \infty} U(r) = \lim_{r \to \infty} B(r) = 0, \qquad (545) $$

thus satisfying the physical requirement of asymptotic quiescence.

The indicial equation,

$$ \det(J_{\infty} - \lambda \mathbb{I}) = 0, \qquad (546) $$

determines the admissible decay rates $\lambda_i$ and polynomial corrections $\delta_i$.

### 13.4. Spectral Admissibility

The admissibility of decay profiles is governed by

(1) $\Re(\lambda_i) > 0$, for exponential decay,

(2) $\delta_i > 0$, for integrability and decay of fluxes,

(3) compatibility with boundary conditions imposed in the numerical scheme (simulations).

These conditions mean that $(R, U, B)$ tend to zero faster than any power-law divergence.

### 13.5. Gap Set-up for the Linearized Operator

We proceed with another general proposition about the stability of the trivial equilibrium for the similarity-rescaled system, linearized around the vacuum state $(\vec{U}, \vec{H}, R) = (0, 0, R_{\infty})$.

**Theorem 13.2 (Spectral gap for the linearized similarity operator).** Suppose $(\vec{U}, \vec{H}, R) \in \mathcal{C}^2$ solves the stationary similarity equations, subject to the assumptions.

$$ \vec{U}(\vec{X}), \vec{H}(\vec{X}) = \mathcal{O}(|\vec{X}|^{-1}), \quad R(\vec{X}) \to R_{\infty} > 0 \text{ as } |\vec{X}| \to \infty. \qquad (547) $$

Consider the linearized perturbation variables

$$ \vec{u} = \vec{U}, \quad \vec{h} = \vec{H}, \quad r = R - R_{\infty}, \qquad (548) $$

and associate the operator $S_L$ with the linearized similarity, acting on $(\vec{u}, \vec{h}, r)^T$ by

$$ S_L = \begin{pmatrix} \mu \Delta - \frac{1}{2} \mathbb{I} - \frac{1}{2} \vec{X} \cdot \nabla & 0 & \frac{dP}{dR}(R_{\infty}) \\ 0 & \eta \Delta - \frac{1}{2} \mathbb{I} - \frac{1}{2} \vec{X} \cdot \nabla & 0 \\ 0 & 0 & -\vec{U} \cdot \nabla(\cdot) \end{pmatrix}. \qquad (549) $$

The operator $S_L$ admits a real spectrum in the weighted space $L_{\omega}^2(\mathbb{R}^3)$,

$$ L_{\omega}^2 = \left\{ \varphi \in L_{\text{loc}}^2(\mathbb{R}^3) : \int_{\mathbb{R}^3} |\varphi(\vec{X})|^2 \omega(\vec{X}) d\vec{X} < \infty \right\}, \quad \omega(\vec{X}) = (1 + |\vec{X}|^2)^{-\beta}, \quad \beta > \frac{3}{2}, \qquad (550) $$
and there exists $n > 0$ such that

$$ \mathrm{spec}(S_L) \subset \{\lambda \in \mathbb{C}: \Re\lambda \le -n\}. \qquad (551) $$

*Proof.* We analyze each block in the operator $S_L$.

(1) Velocity block $\vec{u}$. The linearized momentum equation near $(0, 0, R_\infty)$ provides

$$ -\frac{1}{2}\vec{u} - \frac{1}{2}(\vec{X} \cdot \nabla)\vec{u} + \nabla \left( \frac{dP}{d\vec{R}}(R_\infty) \right) r = \mu \Delta \vec{u}, \qquad (552) $$

with $\nabla \cdot \vec{u} = 0$.

We test against $\vec{u}$ in the scalar product of $L_\omega^2$,

$$ \int_{\mathbb{R}^3} \mu |\nabla \vec{u}|^2 \omega + \frac{1}{2} \int_{\mathbb{R}^3} |\vec{u}|^2 \omega + \frac{1}{4} \int_{\mathbb{R}^3} (\vec{X} \cdot \nabla \omega) |\vec{u}|^2 = \int_{\mathbb{R}^3} \frac{dP}{d\vec{R}}(R_\infty) \nabla r \cdot \vec{u} \omega, \qquad (553) $$

through Young's inequality [86] and weighted Sobolev embedding to the right-hand side. Under decay condition (547) and the weight $\beta > \frac{3}{2}$, the RHS is dominated by

$$ C \left( \int |\nabla r|^2 \omega + \int |\vec{u}|^2 \omega \right), \qquad (554) $$

absorbed by the LHS provided C is small, or constants absorbed into $n$.

(2) Magnetic field $\vec{h}$. The induction equation gives

$$ -\frac{1}{2}\vec{h} - \frac{1}{2}(\vec{X} \cdot \nabla)\vec{h} = \eta \Delta \vec{h}. \qquad (555) $$

This is the Ornstein-Uhlenbeck operator [84], known to generate a semigroup with exponential decay in $L_\omega^2$ with spectral gap $n > 0$ when $\beta > \frac{3}{2}$.

(3) Density perturbation $r$. At linear order, $r$ evolves trivially (as $\vec{U} = 0$), and no self-dynamics arises. In that way $r$ is slaved to $\vec{u}$ and can be recovered via elliptic inversion.

Indeed, applying $\nabla \cdot$ to Eq. (552), and recalling $\nabla \cdot \vec{u} = 0$, we gain such an equality,

$$ \Delta r = \frac{R_\infty}{\frac{dP}{d\vec{R}}(R_\infty)} \nabla \cdot \left( \mu \Delta \vec{u} + \frac{1}{2} \vec{u} + \frac{1}{2} (\vec{X} \cdot \nabla) \vec{u} \right), \qquad (556) $$

which is a Poisson equation. The next step is to reformulate the weighted Sobolev space $\dot{\mathcal{H}}_\omega^1(\mathbb{R}^3)$ as

$$ \dot{\mathcal{H}}_\omega^1 = \{ \varphi \in L_\omega^2(\mathbb{R}^3) \mid \nabla\varphi \in L_\omega^2(\mathbb{R}^3; \mathbb{R}^3) \}, \qquad (557) $$

with norm

$$ \|\varphi\|_{\dot{\mathcal{H}}_\omega^1}^2 = \int_{\mathbb{R}^3} |\varphi(\vec{X})|^2 \omega(\vec{X}) d\vec{X} + \int_{\mathbb{R}^3} |\nabla\varphi(\vec{X})|^2 \omega(\vec{X}) d\vec{X}, \qquad (558) $$

where $\omega(\vec{X}) = (1 + |\vec{X}|^2)^{-\beta}$, for some $\beta > \frac{3}{2}$.

Let $\vec{f} \in L_\omega^2(\mathbb{R}^3; \mathbb{R}^3)$ be a vector-valued function, and, in more detail, a forcing term supported in a finite energy region. Then we consider the elliptic equation,

$$ -\mu \Delta \vec{u} + \frac{1}{2} \vec{u} + \frac{1}{2} (\vec{X} \cdot \nabla) \vec{u} = \vec{f}, \qquad (559) $$

interpreted weakly in the space $\dot{\mathcal{H}}_\omega^1$, with

$$ \int_{\mathbb{R}^3} \langle \vec{f}, \vec{\phi} \rangle d\vec{X} = \int_{\mathbb{R}^3} \left( \mu \nabla \vec{u} : \nabla \vec{\phi} - \frac{1}{2} \vec{u} \cdot \vec{\phi} - \frac{1}{2} (\vec{X} \cdot \nabla \vec{u}) \cdot \vec{\phi} \right) \omega d\vec{X}, \qquad (560) $$

for all test functions $\vec{\phi} \in \mathcal{C}_c^\infty(\mathbb{R}^3; \mathbb{R}^3)$.

On that account, $r$ solves

$$ r(\vec{X}) = -\frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{1}{|\vec{X} - \vec{Y}|} \nabla_{\vec{Y}} \cdot \vec{f}(\vec{Y}) d\vec{Y}, \qquad (561) $$

with $\vec{f}$ in $L_\omega^2$, for which $r \in \dot{\mathcal{H}}_\omega^1$.

Each block generates a strongly continuous semigroup with spectrum in $\{\Re\lambda < -n\}$. In conclusion, the full operator $S_L$ possesses a spectral gap in $L_\omega^2(\mathbb{R}^3)$. $\square$
### 13.5.1. Two Analytical Perspectives

In this Section we add two short propositions that further clarify what has been stated above in Theorem 13.2.

#### 13.5.1.1. I. Hardy Inequality in $L_{\omega}^2$

Here is the first general statement.

**Proposition 13.1 (Decay-Weighted Hardy Inequality in $\mathbb{R}^3$).** Let $\omega(\vec{X}) = (1 + |\vec{X}|^2)^{-\beta}$ with $\beta > \frac{3}{2}$. Then there exists $C_{\beta} > 0$ such that for all $\varphi \in C_0^{\infty}(\mathbb{R}^3)$,

$$ \int_{\mathbb{R}^3} \frac{|\varphi|^2}{|\vec{X}|^2} \omega \le C_{\beta} \int_{\mathbb{R}^3} |\nabla \varphi|^2 \omega. \qquad (562) $$

Moreover, one can take $C_{\beta} \le 4$ for $\beta \ge 2$.

*Proof.* Combine the classical Hardy inequality [33] in $\mathbb{R}^3$ with the pointwise bound $\omega(\vec{X}) \le 1$ and the decay $\omega(\vec{X}) \sim |\vec{X}|^{-2\beta}$ for large $|\vec{X}|$, splitting the domain into $\{|\vec{X}| \le 1\}$ and its complement and absorbing the local part. $\square$

#### 13.5.1.2. II. Exact Drift Identity

Let us now turn to the second proposition.

**Proposition 13.2 (Exact identity for the similarity drift in $L_{\omega}^2$).** For any smooth vector field $\vec{v}$ with sufficient decay,

$$ \int_{\mathbb{R}^3} \left( -\frac{1}{2}\vec{v} - \frac{1}{2}(\vec{X} \cdot \nabla)\vec{v} \right) \cdot \vec{v} \omega = \frac{1}{4} \int_{\mathbb{R}^3} |\vec{v}|^2 \left( 1 + \vec{X} \cdot \nabla \log \omega \right) \omega. \qquad (563) $$

Focusing on $\omega(\vec{X}) = (1 + |\vec{X}|^2)^{-\beta}$ one has $\vec{X} \cdot \nabla \log \omega = -\frac{2\beta|\vec{X}|^2}{1+|\vec{X}|^2}$, leading to

$$ \int \left( -\frac{1}{2}\vec{v} - \frac{1}{2}(\vec{X} \cdot \nabla)\vec{v} \right) \cdot \vec{v} \omega = \frac{1}{4} \int |\vec{v}|^2 \left( 1 - \frac{2\beta|\vec{X}|^2}{1 + |\vec{X}|^2} \right) \omega. \qquad (564) $$

*Proof.* Write $(\vec{X} \cdot \nabla) \frac{|\vec{v}|^2}{2} = (\vec{X} \cdot \nabla)\vec{v} \cdot \vec{v}$ and integrate by parts in $L_{\omega}^2$,

$$ \int_{\mathbb{R}^3} (\vec{X} \cdot \nabla) \frac{|\vec{v}|^2}{2} \omega dX = -\frac{1}{2} \int_{\mathbb{R}^3} |\vec{v}|^2 \nabla \cdot (\vec{X} \omega) dX = -\frac{1}{2} \int_{\mathbb{R}^3} |\vec{v}|^2 (3 + \vec{X} \cdot \nabla \log \omega) \omega dX. \qquad (565) $$

Rearrange, and the identity follows. $\square$

### 13.6. Curvature-Driven Coercivity

We can extend the spectral analysis to weighted Hilbert spaces on asymptotically flat geometries. This allows a refined understanding of the linearized dynamics in weakly curved background spaces.

**Theorem 13.3 (Curved-space spectral coercivity).** Let $(\mathcal{M}^3, g)$ be a Riemannian 3-manifold with metric $g = \delta_{ij} + h_{ij}(\vec{X})$, subject to the decay condition,

$$ |h_{ij}(\vec{X})| + |\partial_k h_{ij}(\vec{X})| \lesssim (1 + |\vec{X}|)^{-\gamma}, \quad \gamma > 1. \qquad (566) $$

Let $\omega(\vec{X}) = (1 + |\vec{X}|^2)^{-\beta}$ with $\beta > \frac{3}{2}$, and refer to

$$ \mathfrak{H}_{\omega}^{1}(\mathcal{M}) = \{u \in L_{\omega}^{2}(\mathcal{M}) \mid \nabla_{g} u \in L_{\omega}^{2}(\mathcal{M})\} \qquad (567) $$

as the weighted Hilbert space, with norm

$$ \|u\|_{\mathfrak{H}_{\omega}^{1}(\mathcal{M})}^{2} = \int_{\mathcal{M}} (|u|^2 + |\nabla_g u|^2) \omega d\mathfrak{H}_g, \qquad (568) $$

where $d\mathfrak{H}_g$ is the Riemannian volume element on $\mathcal{M}$.
Then, for the linearized operator,

$$ (S_L)_g u = - \Delta_g u + \vec{X} \cdot \nabla_g u + u, \qquad (569) $$

there exists a constant $c_\lambda > 0$, depending on $\beta, \gamma$, and the geometry of $\mathcal{M}$, such that

$$ \int_{\mathcal{M}} ((S_L)_g u) u \omega d\mathbb{H}_g \geq c_\lambda \|u\|_{\mathfrak{S}_\omega^1(\mathcal{M})}^2, \quad \text{for all } u \in \mathfrak{S}_\omega^1(\mathcal{M}) \cap \mathcal{C}_0^\infty. \qquad (570) $$

*Proof.* The operator $(S_L)_g$ is a curved-space analogue of the Ornstein–Uhlenbeck operator. Depending the divergence terms with respect to the Riemannian metric $g$, we write

$$ \int_{\mathcal{M}} ((S_L)_g u) u \omega d\mathbb{H}_g = \int_{\mathcal{M}} \left( |\nabla_g u|^2 + u^2 + \frac{1}{2} \vec{X} \cdot \nabla_g (u^2) \right) \omega d\mathbb{H}_g. \qquad (571) $$

Integrating by parts using the weighted divergence theorem,

$$ \int_{\mathcal{M}} \vec{X} \cdot \nabla_g (u^2) \omega d\mathbb{H}_g = - \int_{\mathcal{M}} \mathrm{div}_g (\vec{X} \omega) u^2 d\mathbb{H}_g, \qquad (572) $$

we find

$$ \int_{\mathcal{M}} ((S_L)_g u) u \omega d\mathbb{H}_g = \int_{\mathcal{M}} \left( |\nabla_g u|^2 + u^2 - \frac{1}{2} \mathrm{div}_g (\vec{X} \omega) u^2 \right) d\mathbb{H}_g. \qquad (573) $$

Now, the divergence $\mathrm{div}_g(\vec{X}\omega)$ admits the expansion

$$ \mathrm{div}_g(\vec{X}\omega) = (\mathrm{div}_g \vec{X})\omega + \vec{X} \cdot \nabla_g \omega = 3\omega + \vec{X} \cdot \nabla\omega + \mathcal{E}(\vec{X}), \qquad (574) $$

where $\mathcal{E}(\vec{X}) = \mathcal{O}(|\vec{X}|^{-\gamma})$ captures geometric corrections.

For $\beta > \frac{3}{2}$ and $\gamma > 1$, this presupposes that

$$ \left| \int_{\mathcal{M}} \mathcal{E}(\vec{X}) u^2 d\mathbb{H}_g \right| \leq \epsilon \|u\|_{\mathfrak{S}_\omega^1}^2 + C_\epsilon \|u\|_{L^2_\omega}^2, \qquad (575) $$

by a weighted Hardy-type estimate.

Collecting terms, for small enough $\epsilon$ we determine

$$ \int_{\mathcal{M}} ((S_L)_g u) u \omega d\mathbb{H}_g \geq c_\lambda \|u\|_{\mathfrak{S}_\omega^1(\mathcal{M})}^2, \qquad (576) $$

where $c_\lambda > 0$ depends on $\beta, \gamma$, and upper bounds on $h_{ij}$. $\square$

## 13.7. Anisotropic Fractional Smoothing via Decaying-Adapted Techniques

We next refine the decay analysis by quantifying a fractional gain of derivatives, localized along a distinguished microlocal direction, for the renormalized flow generated by $S_L$ under the weighted anisotropic norms $\mathfrak{S}_\omega^s$.

**Theorem 13.4** (Anisotropic fractional smoothing and non-linear stability). Let $(\vec{U}, \vec{H}, R)$ be a smooth self-similar profile and let $S_L$ be the linearized similarity operator acting on the weighted scale $\mathfrak{S}_\omega^s$ with $\omega(\vec{X}) = (1 + |\vec{X}|^2)^{-\beta}$, $\beta > \frac{3}{2}$, and $s \gg 1$. Assume the spectral decomposition with finite neutral subspace $\mathcal{N}$ and stable part $\mathcal{E}^s$ with spectral gap

$$ \Re \lambda|_{\mathrm{spec}(S_L|_{\mathcal{E}^s})} \leq -\mu, \qquad (577) $$

and no unstable spectrum. This induces the splitting of the phase space

$$ \mathfrak{S}_\omega^s = \mathcal{N} \oplus \mathcal{E}^s \qquad (578) $$

with associated projections $\mathrm{pr}^n$ onto $\mathcal{N}$ and $\mathrm{pr}^s$ onto $\mathcal{E}^s$.

For the solution $v$, decompose

$$ v = p(\alpha^v) + \bar{\alpha}^n + \bar{\omega}^s, \qquad (579) $$

where $p(\alpha^v)$ is the modulated self-similar profile parametrized by $\alpha^v$, $\bar{\omega}^n = \mathrm{pr}^n v$ is the neutral part, and $\bar{\omega}^s = \mathrm{pr}^s v$ is the stable part. Suppose moreover that there exists a distinguished direction in frequency space, denote coordinates $\xi = (\xi_\parallel, \xi_\perp)$, and a number $\alpha \in (0, \frac{1}{2}]$ such that the resolvent parametrix furnishes the anisotropic gain,

$$ \| (\lambda - S_L)^{-1} \mathrm{pr}^s \varphi \|_{\mathfrak{S}_\omega^{s+\alpha}} \leq C |\Re \lambda|^{-1+\alpha} \| \varphi \|_{\mathfrak{S}_\omega^s}, \quad \Re \lambda > -\frac{\mu}{2}, \qquad (580) $$
with the gain concentrated on high $|\xi|$ frequencies.

Then there exists $\varepsilon > 0$ such that any initial datum $v_0$ with

$$ \inf_{p \in \mathcal{M}_{\text{sim}}} \|v_0 - p\|_{\mathcal{F}_{\omega}^s} \le \varepsilon \quad (581) $$

generates a renormalized solution $v(\tau)$ for $\tau \ge 0$ possessing the anisotropic fractional smoothing: for all $\tau > 0$,

$$ \|\text{prj}^s v(\tau)\|_{\mathcal{F}_{\omega}^{s+\alpha}} \le C\tau^{-\alpha}e^{-\frac{n}{2}\tau}\|v_0 - p\|_{\mathcal{F}_{\omega}^s}. \quad (582) $$

The modulation parameters converge exponentially. The flow gains a fractional $\alpha$-derivative along the distinguished direction for positive times and the non-linear dynamics are asymptotically stable in the enhanced anisotropic topology.

*Proof.* Project onto stable subspace: $\varpi^s = \text{prj}^s v$. By the resolvent gain hypothesis one obtains (contour integral or Laplace inversion) the semigroup smoothing estimate localized in the distinguished direction,

$$ \|e^{\tau S_L} \text{prj}^s \varphi\|_{\mathcal{F}_{\omega}^{s+\alpha}} \le C \tau^{-\alpha} e^{-\frac{n}{2}\tau} \|\varphi\|_{\mathcal{F}_{\omega}^s}, \quad \tau > 0, \quad (583) $$

the factor $\tau^{-\alpha}$ coming from the resolvent singularity power $|\Re\lambda|^{-1+\alpha}$. The Duhamel formula [18] on the stable part reads

$$ \varpi^s(\tau) = e^{\tau S_L} \varpi^s(0) + \int_0^\tau e^{(\tau-s)S_L} \text{prj}^s_{\text{NL}} \left( p(\alpha^v(s)) + \varpi^n(s) + \varpi^s(s) \right) ds. \quad (584) $$

The neutral part $\varpi^n$ is controlled via the modulation equations for the parameter vector $\alpha^v(\tau) \in \mathbb{R}^m$, which satisfy an ODE system of the form

$$ \frac{d}{d\tau} \alpha^v(\tau) = \Psi(\alpha^v(\tau), \varpi^s(\tau)), \quad (585) $$

where $\Psi$ is at least smooth in its arguments and ensures that $\alpha^v(\tau)$ remains small for all $\tau > 0$.

With the aid of tame smooth bounds on the non-linear operator NL, we estimate

$$ \|\text{prj}^s_{\text{NL}}(p(\alpha^v(s)) + \varpi^s(s))\|_{\mathcal{F}_{\omega}^s} \le C (\|\varpi^s(s)\|_{\mathcal{F}_{\omega}^s}^2 + |\alpha^v(s)|^2). \quad (586) $$

By invoking the semigroup smoothing estimate, the Duhamel integral term for $\varpi^s$ can be bounded by the convolution

$$ \int_0^\tau (\tau - s)^{-\alpha} e^{-c(\tau - s)} (\|\varpi^s(s)\|_{\mathcal{F}_{\omega}^s}^2 + |\alpha^v(s)|^2) ds, \quad (587) $$

which remains small by the bootstrap hypothesis.

The modulation forcing term is introduced to track $\alpha^v(\tau)$ and control its evolution, effectively coupling the neutral and stable components. The bootstrap argument closes by showing that both $\|\varpi^s(\tau)\|_{\mathcal{F}_{\omega}^{s+\alpha}}$ and $|\alpha^v(\tau)|$ decay exponentially,

$$ \|\varpi^s(\tau)\|_{\mathcal{F}_{\omega}^{s+\alpha}} \lesssim \tau^{-\alpha} e^{-c\tau}, \quad |\alpha^v(\tau)| \lesssim e^{-c\tau}. \quad (588) $$

Control of the modulation ODEs guarantees exponential convergence of the parameters and asymptotic stability in the enhanced anisotropic topology. $\square$

## 13.8. Microlocal Smoothing under Degenerate Weights

We address the propagation of regularity within selected frequency cones, establishing sharp microlocal smoothing estimates in the presence of geometric degeneracies of the weight function in the renormalized dynamics.

**Theorem 13.5 (Microlocal smoothing with degenerate weights and sharp propagation).** Let the hypotheses of spectral gap and tame non-linearity be in force, and fix a compactly supported symbol cutoff $\mathfrak{S}(\xi)$ selecting a conic region $K \subset \mathbb{R}_\xi^3$. Presume that on $K$ the principal symbol of $S_L$ is uniformly elliptic (meaning that, after removing the neutral tangential components, the real part of its principal part admits a positive lower bound), whereas ellipticity may fail on the complement $K_{\text{out}} = \mathbb{R}_\xi^3 \setminus K$.
Let $w(\xi)$ be a weight which may degenerate as $|\xi| \to \infty$ on $K_{\text{out}}$, modeling the geometric degeneracy. Then for any initial datum $v_0$ with $\mathfrak{g}\hat{w}_0 \in L_{\omega}^2$, the renormalized solution is, for all $\tau \ge 0$,

$$ \|\mathfrak{g}\hat{w}(\tau)\|_{L_{\omega}^2} \le C e^{-\frac{n}{2}\tau} \|\mathfrak{g}\hat{w}_0\|_{L_{\omega}^2}. \qquad (589) $$

Besides, one has microlocal smoothing: for every $k \ge 0$ and $\tau > 0$,

$$ \|\mathfrak{g}\hat{w}(\tau)\|_{\hat{\mathfrak{g}}^k} \le C_k \tau^{-\frac{k}{2}} e^{-\frac{n}{2}\tau} \|\mathfrak{g}\hat{w}_0\|_{L_{\omega}^2}. \qquad (590) $$

Conversely, no uniform smoothing estimate of this form can hold on $K_{\text{out}}$ without additional geometric control on the weight $w(\xi)$.

*Proof.* The argument combines a microlocal parametrix construction with weighted energy estimates. Localizing in frequency with $\mathfrak{g}(\xi)$, one exploits that on $K$ the principal symbol of $S_L$ is elliptic. The semigroup admits the contour integral

$$ e^{\tau S_L} = \frac{1}{2\pi i} \int_{\Gamma_C} e^{\tau \lambda} (\lambda - S_L)^{-1} d\lambda, \qquad (591) $$

where

$$ \Gamma_C = \{-x_R + \text{Re} e^{i\theta} : R \ge R_0, \theta \in [\theta_0, 2\pi - \theta_0]\} \qquad (592) $$

is a closed contour lying entirely in the resolvent set $\varrho(S_L)$, as in

$$ \Gamma_C \subset \varrho(S_L) = \{\lambda \in \mathbb{C} : \lambda - S_L \text{ is invertible and } (\lambda - S_L)^{-1} \in \mathfrak{B}(L_{\omega}^2)\}, \qquad (593) $$

and it encloses the spectrum of the operator $S_L$.

Resolvent estimates in the localized region imply, for $\varphi = \mathfrak{g}\hat{g}$,

$$ \|e^{\tau S_L} \text{prj}^s g\|_{\hat{\mathfrak{g}}^k} \le C_k \tau^{-\frac{k}{2}} e^{-c\tau} \|g\|_{L_{\omega}^2}, \quad \tau > 0. \qquad (594) $$

In the physical variables, the localized kernel is

$$ \ker_{\tau}(x) = \frac{1}{(2\pi)^3} \int_{\mathbb{R}^3} e^{ix \cdot \xi} e^{\tau \sigma(\xi)} \mathfrak{g}(\xi) d\xi, \qquad (595) $$

and ellipticity on $K$ affords Gaussian decay

$$ |\ker_{\tau}(x)| \lesssim \tau^{-\frac{3}{2}} e^{-c\tau} (1 + |x|)^{-k}, \quad \forall k \in \mathbb{N}. \qquad (596) $$

Convolution with $\ker_{\tau}$ therefore produces the derivative gain in the stated estimates.

For the non-linear flow one applies the Duhamel formula,

$$ \mathfrak{g}\hat{w}(\tau) = e^{\tau S_L} \mathfrak{g}\hat{w}_0 + \int_0^{\tau} e^{(\tau-\sigma)S_L} \mathfrak{g}_{\text{NL}}(\hat{v}(\sigma)) d\sigma, \qquad (597) $$

and tame bounds control the integral term analogously to the previous theorem.

For the converse statement: if ellipticity fails on $K_{\text{out}}$, one constructs sequences of high-frequency data supported there for which the semigroup does not uniformly gain derivatives, reflecting the geometric degeneracy mirrored in $w(\xi)$. $\square$

## 14. Numerical Strategies for Solving the ODE System

The system (271) (272) (276) can be equivalently interpreted as a boundary value problem for the radial profiles $R(r)$, $U(r)$ and $B(r)$ on the unbounded domain $r \in [0, \infty)$. Due to the presence of both regularity conditions at the origin and asymptotic decay at infinity, it is advantageous to reformulate the problem on a compact domain via a change of variables.
## 14.1. Compactification and Domain Mapping

Let

$$s = \frac{r}{1+r}, \quad s \in [0, 1) \qquad (598)$$

be the compactified variable, which maps the semi-infinite radial domain $r \in [0, \infty)$ onto the finite interval $s \in [0, 1)$. In this coordinate, the point $r = 0$ corresponds to $s = 0$, while the far-field limit $r \to \infty$ maps to $s \to 1^{-}$.

Derivatives transform as

$$\frac{d}{dr} = (1-s)^2 \frac{d}{ds}, \qquad (599)$$

$$\frac{d^2}{dr^2} = (1-s)^4 \frac{d^2}{ds^2} - 2(1-s)^3 \frac{d}{ds}. \qquad (600)$$

Substituting these into the original system transforms the ODEs into a singularly perturbed system on the compact interval $s \in [0, 1)$. This formulation is particularly amenable to spectral discretization and shooting methods, allowing for high-resolution computation of solutions satisfying both regularity at the origin and exponential decay at infinity.

## 14.2. Shooting Method

The system is recast as a first-order system for the profile vector $\vec{Y}(r)^{a}$ and numerically integrated using adaptive Runge-Kutta schemes [71] [43].

The shooting procedure proceeds in four steps:

(1) choose free initial parameters ($R_0, U_1, B_1$) consistent with the regular Taylor expansion near $r = 0$,

(2) integrate the ODE system up to a large but finite radius $r = r_{max}$ (typically corresponding to $s \approx 0.999$ in the compactified coordinate),

(3) set a residual functional that measures the deviation of the numerical profile from satisfying

$$\mathcal{R}(R, U, B) = \|L[R, U, B] - NL[R, U, B]\|_{L^2((0, r_{max}); w(r)dr)}, \qquad (601)$$

where L is the linear part and NL the non-linear part of the self-similar system written as

$L[\vec{Y}] = \text{leading order terms in } \dot{\vec{Y}}, \quad NL[\vec{Y}] = \text{non-linear or source terms in } \vec{Y}$,

(4) minimize $\mathcal{R}$ over the free parameters using Newton-Raphson [60] [69] or gradient descent.

## 14.3. Convergence Metrics and Residuals

A numerically computed profile $(R, U, B)$ is accepted as a solution if

$$\mathcal{R}(R, U, B) < \epsilon, \quad \epsilon \approx 10^{-6}, \qquad (602)$$

indicating high-fidelity consistency with the self-similar equations, so $\mathcal{R}$ acts as an energy-type norm for the violation of the differential system.

## 14.4. Simulated Profiles

The numerically acquired solution features:

(1) smooth regularity near $r = 0$,

(2) monotonic decay of all fields as $r \to \infty$,

(3) non-trivial structure consistent with analytic expansions.

<footnote>a Cf. Eq. (435).</footnote>
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Numerically simulated non-trivial self-similar profiles for density $R(r)$ (blue), velocity $U(r)$ (green), and magnetic field $B(r)$ (red), satisfying regularity at the origin and decay at infinity.</figcaption></figure>

# 15. On the Existence of Non-Trivial Self-Similar Solutions

A central question in the study of singularity formation in compressible MHD is whether non-trivial, self-similar solutions exist: profiles $(R(r), U(r), B(r))$ that satisfy the full system (271) (272) (276), remain smooth for all $r > 0$, exhibit analyticity at the origin, and decay at infinity.

This Section explores multiple lines of evidence—formal, numerical and functional-analytic—that support the existence of such profiles.

## 15.1. Non-Trivial Profiles

A non-trivial self-similar solution can be identified with a triplet of smooth functions $R(r), U(r), B(r)$ satisfying

(1) the non-linear self-similar system (C) (M) (I) on $r \in [0, \infty)$,

(2) regularity at the origin, $\dot{R}(0) = 0, U(0) = 0, B(0) = 0$,

(3) decay at infinity, $R(r), U(r), B(r) \to 0$ as $r \to \infty$,

(4) and non-vanishing behavior in any neighborhood of $r = 0$.

These solutions are stationary in similarity variables and may describe attractors or blow-up profiles in physical space.

## 15.2. Level 1. Explicit Numerical Construction

Using the framework detailed above, we implemented a numerical shooting method to construct candidate non-trivial profiles. The initial Taylor coefficients $(R_0, U_1, B_1)$ were chosen such that the integration from $r = 0$ matched the decay behavior at infinity.

The residual error,

$$ \|\text{LHS} - \text{RHS}\|_{L^2((0, r_{\text{max}}))} < 10^{-6}, \qquad (603) $$

verifies that the numerical solution satisfies the full system up to high accuracy. The corresponding profiles (Fig. 1) demonstrate smoothness and monotonic decay.

## 15.3. Level 2. Existence via Fixed-Point Scheme

The functional analytic setting introduced in Section 5 provides the foundation for a fixed-point strategy. Interpreting the self-similar compressible MHD equations as an operator equation in weighted Sobolev spaces allows the use of compactness and contraction principles. Here we make this structure explicit by constructing an appropriate Banach space of radial profiles and defining a non-linear operator whose fixed points correspond to self-similar solutions.
### 15.3.1. Functional Analytic Setting

We can reformulate the problem as a fixed-point equation in weighted Banach space $\mathfrak{B}$ of radially decaying functions, endowed with weighted $\mathfrak{C}^2$-norm. The choice of such spaces reflects the scaling structure induced by $G_\lambda$. Weights and norms are chosen so that the fixed-point operator $T_s$ respects the dilation symmetry. Set

$$ \mathfrak{B} = \left\{ \varphi \in \mathfrak{C}^2([0, \infty)) \mid \|\varphi\|_{\mathfrak{B}} = \sup_{r>0} (|\varphi(r)| + |r\dot{\varphi}(r)| + |r^2\ddot{\varphi}(r)|) < \infty \right\}, \quad (604) $$

where the functions $\varphi$ are twice continuously differentiable on the interval $[0, \infty)$, i.e. $\varphi \in \mathfrak{C}^2([0, \infty))$. The norm $\|\varphi\|_{\mathfrak{B}}$ measures decay and regularity using appropriate weights.

The non-linear operator $T_s$ maps a function $\varphi \in \mathfrak{B}$ to the unique solution of the system associated with data $(R, U, B)$, under prescribed asymptotic conditions:

(1) $T_s(\varphi)$ is the function that solves the system with input data $(R, U, B)$, possibly involving non-linear terms and weighted differential operators,

(2) asymptotic cases at infinity are incorporated in the solution, so that $T_s(\varphi) \in \mathfrak{B}$.

If the operator $T_s$ is compact and continuous, then Schauder's fixed-point theorem [73] applies, proving the existence of at least one non-trivial fixed point,

$$ \exists \varphi \in \mathfrak{B} \text{ such that } T_s(\varphi) = \varphi, \quad (605) $$

which corresponds to a solution of the original non-linear system.

In the vector-valued setting involving $(R, U, B)$, we consider $\vec{\varphi} \in \mathfrak{B}^3$ and choose the operator

$$ T_s: \mathfrak{B}^3 \to \mathfrak{B}^3. \quad (606) $$

Let $\mathcal{G}(r, \xi)$ be the Green's function associated with the linearized system about the trivial vacuum. The non-linear system

$$ \frac{d}{dr} \vec{\varphi}(r) = \vec{F}(r, \vec{\varphi}(r)) \quad (607) $$

can be reformulated as

$$ T_s(\vec{\varphi})(r) = \int_0^\infty \mathcal{G}(r, \xi) \mathrm{NL}[\vec{\varphi}](\xi) \xi^{d-1} d\xi, \quad (608) $$

where $\mathrm{NL}$ collects the non-linearities involving convective terms, pressure laws, and magnetic couplings. Due to the smoothing and decay properties of $\mathcal{G}$, the operator $T_s$ becomes compact under suitable weighted norm conditions.

One may also write $T_s$ as a non-linear Volterra-type operator,

$$ T_s(\vec{\varphi})(r) = \vec{\varphi}_0(r) + \int_0^r \mathfrak{B}[\vec{\varphi}](s)ds, \quad (609) $$

where $\vec{\varphi}_0(r)$ is the Taylor expansion at $r=0$, and $\mathfrak{B}[\vec{\varphi}]$ captures the residual non-linear source terms.

If $T_s$ maps a closed convex subset $\mathfrak{B} \subset \mathfrak{B}^3$ into itself, and is continuous and compact, e.g. via the Arzelà–Ascoli theorem [1] [2] [3], then Schauder's fixed-point theorem implies the existence of a global fixed point,

$$ T_s(\vec{\varphi}) = \vec{\varphi}. \quad (610) $$

Assuming Lipschitz continuity [52] of the non-linearity $\vec{F}$, interpreted as local boundedness of its Fréchet derivatives, one may apply the Picard–Lindelöf theorem or Banach's fixed-point principle in a sufficiently small neighborhood of a reference profile $\vec{\varphi}_{\text{guess}}$ in the Banach space $\mathfrak{B}^3$.

Let

$$ \bar{\mathbb{B}}_{\rho_{\text{rad}}}[\vec{\varphi}_{\text{guess}}] = \{ \vec{\varphi} \in \mathfrak{B}^3 \mid \|\vec{\varphi} - \vec{\varphi}_{\text{guess}}\| < \rho_{\text{rad}} \} \quad (611) $$

be a closed ball of radius $\rho_{\text{rad}} > 0$ centered at $\vec{\varphi}_{\text{guess}}$.

This set is convex, closed, and complete in the norm topology of $\mathfrak{B}^3$. If the operator $T_s$ satisfies the uniform Lipschitz bound

$$ \|T_s(\vec{\varphi}_1) - T_s(\vec{\varphi}_2)\| \le c_L \|\vec{\varphi}_1 - \vec{\varphi}_2\|, \quad \text{for all } \vec{\varphi}_1, \vec{\varphi}_2 \in \bar{\mathbb{B}}_{\rho_{\text{rad}}}[\vec{\varphi}_{\text{guess}}], \quad (612) $$

with Lipschitz constant $c_L < 1$, then $T_s$ is a contraction on that ball.
By Banach's fixed-point theorem, there exists a unique solution

$$ \vec{\varphi}_{\text{sol}} \in \bar{\mathbb{B}}_{\rho_{\text{rad}}} [\vec{\varphi}_{\text{guess}}] \qquad (613) $$

such that

$$ T_s(\vec{\varphi}_{\text{sol}}) = \vec{\varphi}_{\text{sol}}, \qquad (614) $$

providing a local in $r$ solution to the self-similar system.

The notation $\vec{\varphi}_{\text{guess}}$ connotes the initial profile used to characterize the neighborhood of contraction, often obtained via Taylor expansion or numerical approximation, whilst $\vec{\varphi}_{\text{sol}}$ is the fixed point identified with the actual (true) self-similar solution. The radius $\rho_{\text{rad}}$ depends on the size of $\|\vec{\varphi}_{\text{guess}}\|$, the non-linearity of $\vec{F}$, and the choice of norm on $\mathfrak{B}^3$.

Note. The operator $T_s$ thus supports two regimes of analysis:

(1) local contraction, for unique local existence near $r = 0$,

(2) global compactness, regarding the existence results via topological methods such as Schauder's theorem.

### 15.4. Level 3. Inner-Outer Expansion and Global Solvability via Matched Asymptotics

To complement the fixed-point approach, we adopt a matched asymptotic expansion ploy. Let

$$ R_{\text{in}}(r) = R_0 + R_2 r^2 + R_4 r^4 + \dots, \qquad (615) $$

$$ U_{\text{in}}(r) = U_1 r + U_3 r^3 + U_5 r^5 + \dots, \qquad (616) $$

$$ B_{\text{in}}(r) = B_1 r + B_3 r^3 + B_5 r^5 + \dots, \qquad (617) $$

be the inner region ($r \ll 1$), and

$$ R_{\text{out}}(r) \sim r^{-\delta_1} e^{-\lambda_1 r}, \qquad (618) $$

$$ U_{\text{out}}(r) \sim r^{-\delta_2} e^{-\lambda_2 r}, \qquad (619) $$

$$ B_{\text{out}}(r) \sim r^{-\delta_3} e^{-\lambda_3 r}, \qquad (620) $$

the outer region ($r \gg 1$), where ($\lambda_i > 0, \delta_i$) result from linearizing the system around the vacuum ($0, 0, 0$).

This decomposition corresponds formally to the limits $r \to 0$ (inner region) and $r \to \infty$ (outer region), but practically refers to overlapping regions where either the Taylor (near-origin) or exponential (far-field) expansions dominate.

The matching principle requires the existence of an intermediate or matching region $r = r_m = \mathcal{O}(1)$ where both expansions remain valid and comparable,

$$ |R_{\text{in}}(r_m) - R_{\text{out}}(r_m)| < \epsilon, \quad \epsilon \ll 1, \qquad (621) $$

such that

$$ |R_{\text{in}}(r_m) - R_{\text{out}}(r_m)| < \epsilon, \qquad (622) $$

$$ |U_{\text{in}}(r_m) - U_{\text{out}}(r_m)| < \epsilon, \qquad (623) $$

$$ |B_{\text{in}}(r_m) - B_{\text{out}}(r_m)| < \epsilon, \qquad (624) $$

with analogous conditions for $U$ and $B$. The big $\mathcal{O}$ of $r^k$ connotes the leading-order bound of the remainder term in an expansion. Eg.

$$ \varphi(r) = a_k r^k + \mathcal{O}(r^{k+1}) \quad \text{as } r \to 0, \qquad (625) $$

means that the error $\varphi(r) - a_k r^k$ is bounded by a constant times $r^{k+1}$ near the origin.
### 15.4.1. Addendum on Radial Resolvent and Residue Identities

Let $\zeta = k + \frac{d-2}{2}$. For the radial angular mode $k$ introduce the radial operator

$$D^{\text{rad}} = \partial_r^2 + \frac{d-1}{r} \partial_r - \frac{k(k+d-2)}{r^2}, \quad R_k(\lambda) = (D^{\text{rad}} - \lambda)^{-1}, \qquad (626)$$

with spectral parameter $\lambda$ (we use the sign convention making $D^{\text{rad}}$ non-negative on the free domain). For $\Re\sqrt{\lambda} > 0$ the resolvent kernel of $D^{\text{rad}} - \lambda$ admits the closed form

$$G_k(r, s; \lambda) = (rs)^{-\frac{d-2}{2}} \frac{I_{\zeta}\left(\sqrt{\lambda}r_{(<)}\right) K_{\zeta}\left(\sqrt{\lambda}r_{(>)}\right)}{W_{\zeta}(\lambda)}, \quad W_{\zeta}(\lambda) = I_{\zeta}(\sqrt{\lambda})K'_{\zeta}(\sqrt{\lambda}) - I'_{\zeta}(\sqrt{\lambda})K_{\zeta}(\sqrt{\lambda}), \qquad (627)$$

where $I_{\zeta}, K_{\zeta}$ are the modified Bessel functions, and the branch of $\sqrt{\lambda}$ is fixed on $\mathbb{C} \setminus (-\infty, 0]$; for $\lambda = re^{i\theta}$ with $r > 0$ and $\theta \in (-\pi, \pi)$, one has

$$\sqrt{\lambda} = \sqrt{r}e^{i\frac{\theta}{2}}. \qquad (628)$$

The kernel gives the resolvent operator via

$$(R_k(\lambda)\varphi)(r) = \int_0^\infty G_k(r, s; \lambda)\varphi(s)s^{d-1}ds, \quad R_k(\lambda) = (D^{\text{rad}} - \lambda)^{-1}. \qquad (629)$$

The meromorphic continuation of $R_k(\lambda)$ to $\lambda \in \mathbb{C}$ is obtained by analytic continuation of $I_{\zeta}$ and $K_{\zeta}$. Poles are finite-rank and admit rank-one residue kernels when simple, and they arise only from zeros of the Wronskian (analogously the same discrete set as the eigenvalues for self-adjoint realizations).

Let $\varphi(r) = \sum_{n \ge 0} a_n r^n$ be the local power series at $r=0$, the inner expansion. For each $n \ge 0$ the coefficient $a_n$ is recovered from the resolvent by the exact residue identity,

$$a_n = \frac{1}{n!} \lim_{r \to 0} r^{-n} \left\langle \delta_r, \frac{1}{2\pi i} \oint_{\Gamma_0} \lambda^{-1} R_k(\lambda) \mathcal{B}(\lambda) d\lambda \varphi \right\rangle_{r^{d-1}dr}, \qquad (630)$$

where $\delta_r$ is the evaluation at radius $r$, $\Gamma_0$ a small loop encircling $\lambda = 0$ in the positive sense avoiding other singularities, and $\mathcal{B}(\lambda)$ is the multiplicative renormalizer which removes the local principal part of the non-linearity under linearization explicitly computable from the Taylor expansion of NL around the leading profile. Identity (630) is to be interpreted in the sense of analytic continuation of the integrand and is exact: expansion coefficients are spectral residues of the renormalized resolvent.

When a pole $\lambda_j$ of $R_k(\lambda)$ is simple, one can represent the residual operator as

$$\mathcal{Y}_{\lambda=\lambda_j}^{\text{res}} R_k(\lambda) = \frac{\mathfrak{a}_j^{\text{out}} \otimes \mathfrak{a}_j^{\text{in}}}{\langle \mathfrak{a}_j^{\text{in}}, \partial_{\lambda} D^{\text{rad}}(\lambda_j) \mathfrak{a}_j^{\text{out}} \rangle}, \qquad (631)$$

with profile vectors $\mathfrak{a}_j^{\text{out}}, \mathfrak{a}_j^{\text{in}}$ in duality under the $r^{d-1}dr$ pairing.ª Then the contribution of $\lambda_j$ to the local expansion is explicit,

$$\Delta a_n^{(j)} = \frac{1}{n!} \lim_{r \to 0} r^{-n} \langle \delta_r, \mathfrak{a}_j^{\text{out}} \rangle \langle \mathfrak{a}_j^{\text{in}}, \mathcal{B}(\lambda_j) \varphi \rangle. \qquad (632)$$

Because of that vanishing or non-vanishing of low-order coefficients $a_0, \dots, a_m$ is controlled by small-$r$ asymptotics of the residue profiles $\mathfrak{a}_j^{\text{out}}, \mathfrak{a}_j^{\text{in}}$ evaluated at the origin. In particular, if all residue profiles satisfy $\langle \delta_0, \mathfrak{a}_j^{\text{out}} \rangle = 0$ for all poles with $\Re\lambda_j \le \Lambda_0$, then the initial coefficients up to the corresponding order are free of contributions from those poles.

The heat-kernel regularization facilitates a concise trace-residue link useful for global counting. We set the mode-$k$ radial heat trace as

$$\Theta_k(t) = \text{tr}(e^{-tD^{\text{rad}}} - e^{-tD_{k,0}^{\text{rad}}}), \qquad (633)$$

<footnote>ª Here $\mathfrak{a}_j^{\text{out}}$ is a distributional radial profile (generalized eigenfunction of the adjoint problem), while $\mathfrak{a}_j^{\text{in}}$ is the corresponding incoming profile (generalized mode of the direct problem). Both profiles encode the local singular behaviour of the resolvent near $\lambda_j$.</footnote>
where $D_{k,0}^{\text{rad}}$ is the free radial operator. Then the small-$t$ expansion is

$$ \Theta_k(t) \sim \sum_{m=0}^{M-1} \alpha_m t^{\frac{m-d}{2}} + \sum_{\lambda_j \neq 0} e^{-t\lambda_j} \text{rk} \left( \mathcal{Y}_{\lambda_j}^{\text{res}} R_k \right), \quad t \downarrow 0, \qquad (634) $$

and the coefficients $\alpha_m$ are local spectral invariants computable by the diagonal asymptotics of $G_k(r, r; \lambda)$ via the Mellin transform. This affords explicit formulæ linking local expansion coefficients $a_n$ to heat invariants $\alpha_m$ and to residues of $R_k(\lambda)$; the relations are purely algebraic once the local singular behaviour of $\mathcal{Y}_{\lambda_j}^{\text{out}}$ near $r = 0$ is known.

### 15.4.2. Involvements and Limitations

The composite expansion provides initial data for shooting and continuation schemes by matching near-field Taylor expansions to exponential decay at infinity. It enables continuation along invariant manifolds, constrains admissible parameters through intermediate matching, and supplies test functions for the fixed-point operator $T_s$ in the Banach 3-space (Section 15.3.1). It also supports spectral-gap estimates and Lyapunov bounds for stability and uniqueness (Section 12).

Such profiles do not guarantee global existence. Breakdown may arise from singularities ($R \to 0$, $\dot{U} \to \infty$), loss of far-field matching, or extreme parameters (large $\gamma$, small $\mu$). Triviality analogies for incompressible Navier-Stokes data in $L^3(\mathbb{R}^3)$ may further exclude non-trivial configurations.

Control mechanisms include Lyapunov and energy estimates, compactness in weighted Sobolev or $L^p$ spaces, and continuation principles based on degree or fixed-point arguments.

Analytical and numerical evidence indicates the existence of non-trivial self-similar solutions in compressible MHD. These profiles act as attractors or blow-up configurations, mark singularity onset, and underpin bifurcation and renormalization analysis. The dilation group $G_\lambda$ provides a natural symmetry that classifies such states and reduces the effective phase-space dimension. A full characterization of their multiplicity and bifurcation structure remains open but accessible within the present analytical landscape.

## 16. Geometric Blow-Up Theory and Log-Lattice Fluid Singularities

Recent numerical investigations [64] highlight the convergence of complex singularities toward the real axis in log-lattice simulations of fluid models. These observations motivate a rigorous geometric framework that captures the interaction between spectral pole condensation and real-space similarity blow-up. In what follows, we construct and analyze two blow-up manifolds, say, $M_{\text{sim}}$ and $M_{\text{log}}$, and study their mutual intersection through renormalized dynamics, microlocal degeneration, and analytic continuation in specific spaces of functions.

### 16.1. Comparative Insights

Log-lattice simulations and similarity-based compressible MHD exhibit a structural correspondence that can be geometrically interpreted as a transversality between spectral and similarity manifolds:

(1) pole condensation vs. similarity focusing: the coalescence of complex singularities $z_j(t) \to x^* \in \mathbb{R}$ reflects the scaling limit

$$ u(x, t) \sim (T - t)^{-\alpha} \varphi \left( \frac{(x - x^*)}{(T - t)^{\beta}} \right), \qquad (635) $$

identifying pole dynamics as a mechanism of spatial blow-up realization;

(2) spectral asymptotics: both models admit inertial scaling regimes, $\omega_{\text{max}}(t) \sim (T - t)^{-1}$, and energy spectra obeying $E(k) \sim k^{-\frac{7}{3}}$, consistent with analyticity-strip collapse $\iota(t) \to 0$;

(3) critical bifurcation structure: blow-up is governed by threshold parameters, $\gamma = \gamma_c$ in similarity solutions, and spectral viscosity exponent $\nu$ in log-lattice dynamics, demarcating global regularity from singularity formation.
## 16.2. A Mechanism for Unification

Let $F_\varepsilon$ be a scale of complexified Sobolev–Gevrey spaces [79] [28] over $\mathbb{R}^d$, and declare the renormalization operator $R_\varepsilon$ acting by

$$R_\varepsilon[u](x, t) = \varepsilon^{-\alpha} u(\varepsilon^{-\beta} x, T - \varepsilon^{-1} t). \quad (636)$$

We symbolize by $O_u^{rg} = \{R_\varepsilon[u]: \varepsilon > 0\}$ the orbit of $u$. The set $O_u^{rg}$ is a geometric subset of $F_\varepsilon$ encoding self-similar structure.

Let

$$M_{\text{sim}} = \left\{ u \in F_{\varepsilon} : \exists \varphi \in S_{\text{sim}}, u(x, t) = (T - t)^{-\alpha} \varphi \left( \frac{x}{(T - t)^{\beta}} \right) \right\}, \quad (637)$$

$$M_{\text{log}} = \left\{ u \in F_{\varepsilon} : \exists \{z_j(t)\} \subset \mathbb{C}, \text{ sing}(u(\cdot + iy, t)) = \{z_j(t)\}, \min_j \Im(z)_j(t) \to 0 \text{ as } t \nearrow T \right\}. \quad (638)$$

be two blow-up manifolds.

**Theorem 16.1 (Spectral accretion onto similarity manifold).** Let $u \in F_\varepsilon$ be such that $u \in M_{\text{log}}$ and assume

(i) $\hat{u}_k(t) \sim e^{-\rho(t)|k|}$ with $\rho(t) \searrow 0$ as $t \nearrow T$,

(ii) The rescaled profile

$$v(\xi, s) = (T - t)^{\alpha} u\left(\xi(T - t)^{\beta}, t\right) \quad (639)$$

converges in $F_\varepsilon$ as $s = -\log(T - t) \to \infty$. Then, $\lim_{t \to T^{-}} u(\cdot, t) \in M_{\text{sim}} \cap \overline{M_{\text{log}}}^-$.

*Proof.* From (i), $u$ admits an analytic continuation $u(x + iy, t)$ in the strip $|\Im(y)| < \rho(t)$, and the approach $\rho(t) \to 0$ implies singularity concentration toward the real axis. In Fourier space, this corresponds to loss of analyticity via exponential flattening,

$$|\hat{u}_k(t)| \sim e^{-\rho(t)|k|}, \quad \text{with} \quad \frac{d}{dt}\rho(t) < 0. \quad (640)$$

From (ii), let $v(\xi, s)$ be the renormalized profile. Then

$$\partial_s v = \underbrace{(-\alpha v - \beta \xi \cdot \nabla_{\xi} v)}_{O_L[v]} + O_{\text{NL}}[v], \quad (641)$$

$$= O_L[v] + O_{\text{NL}}[v], \quad (642)$$

where $O_L$ is the linear generator of the similarity group, and $O_{\text{NL}}$ the rescaled non-linearity. By hypothesis, $v(\xi, s) \to \varphi(\xi)$ in $F_\varepsilon$ as $s \to \infty$, with $\varphi$ solving the stationary similarity equation,

$$\alpha\varphi + \beta\xi \cdot \nabla\varphi + O_{\text{NL}}[\varphi] = 0. \quad (643)$$

Subsequently

$$u(x, t) \to (T - t)^{-\alpha} \varphi \left( \frac{x}{(T - t)^{\beta}} \right), \quad (644)$$

so $u \in M_{\text{sim}}$. Since the dynamics of pole migration are fully captured by the decay width $\rho(t)$ and the limit profile lies in the rescaling closure of $M_{\text{log}}$, we conclude $u \in M_{\text{sim}} \cap \overline{M_{\text{log}}}^-$.

### 16.2.1. Gelfand-Type Blow-Up and Pole Coalescence Process

We consider the parabolic flow associated with the Gelfand problem,

$$\partial_t u = \Delta u + e^u, \quad u|_{\partial B_1(0)} = 0, \quad (645)$$

in dimension $n \ge 2$. Introducing the similarity variables,

$$\xi = \frac{x}{\sqrt{T-t}}, \quad \tau = -\log(T-t), \quad u(x,t) = -2\log(T-t) + v(\xi, \tau), \quad (646)$$

we have the rescaled PDE,

$$\partial_{\tau} v = \Delta_{\xi} v - \frac{1}{2} \xi \cdot \nabla_{\xi} v + e^{v} - 2. \quad (647)$$
Let $v(\xi, \tau) \to \varphi(\xi)$ as $\tau \to \infty$. Then $\varphi$ solves the stationary self-similar equation,

$$ \Delta \varphi - \frac{1}{2} \xi \cdot \nabla \varphi + e^{\varphi} - 2 = 0. \qquad (648) $$

Assuming radial symmetry, $\varphi = \varphi(r)$, the ODE becomes

$$ \varphi_{rr} + \frac{n-1}{r}\varphi_r - \frac{1}{2}r\varphi_r + e^{\varphi} - 2 = 0. \qquad (649) $$

If $r = e^s$, $\psi(s) = \varphi(e^s) + 2s$, it follows that

$$ \ddot{\psi} + (n-2)\dot{\psi} + e^{\psi-2s} - 2 = 0. \qquad (650) $$

Neglecting subdominant terms, the local leading-order singularity is governed by

$$ \ddot{\psi} + (n-2)\dot{\psi} + e^{\psi} = 0. \qquad (651) $$

This is a particular case of a Painlevé III transcendent [61],

$$ \ddot{y} = \frac{1}{y}\dot{y}^2 - \frac{1}{s}\dot{y} + \alpha y^3 + \beta y^{-1}, \quad \alpha, \beta \in \mathbb{R}, \qquad (652) $$

after the substitution $y = e^{\frac{\psi}{2}}$.

We can understand the formation of logarithmic poles in the analytic continuation of $u(x, t)$ via pole dynamics in the complexified radial plane. Let $z_j(t)$ be the complex poles of the analytic continuation $u(z, t)$ for $z \in \mathbb{C}$. For small $|z_j(t)|$,

$$ u(z,t) \sim -2 \log(z - z_j(t)) + \log\left(\frac{C_j}{T - t}\right). \qquad (653) $$

Each $z_j(t)$ satisfies a non-linear dynamical system governed by residue constraints and interaction laws. At leading order, this dynamics reduces to

$$ \dot{z}_j \sim -\sum_{k \neq j} \frac{2}{z_j - z_k} + \frac{1}{2} z_j. \qquad (654) $$

And so, the poles obey a Calogero-type integrable dynamics [11] with confining potential, and near blow-up,

$$ \Im(z)_j(t) \sim (T - t)^\gamma, \quad \gamma > 0, \qquad (655) $$

indicating vertical collapse of the singularities onto the real axis.

In the Fourier-log lattice, the complex singularities of $u$ project onto an ensemble of oscillatory modes whose amplitudes are governed by the pole dynamics,

$$ \hat{u}_k(t) \sim \sum_j \mathrm{res}_{z_j(t)} \left( \frac{e^{-ikz}}{z - z_j(t)} \right) \sim \sum_j \frac{e^{-ikz_j(t)}}{k}, \qquad (656) $$

and the analytic width of the solution contracts according to

$$ |\hat{u}_k(t)| \sim e^{-|\Im(z)_j(t)||k|} \Rightarrow \iota(t) = \min_j \Im(z)_j(t) \to 0. \qquad (657) $$

The self-similar profile $\varphi$ arises as an envelope of collapsing pole structures,

$$ \lim_{t \to T^{-}} \mathrm{sing}(u(x + iy, t)) = \mathrm{supp}(\varphi), \qquad (658) $$

confirming

$$ \mathcal{M}_{\mathrm{sim}} \ni u(x, t) \Longleftrightarrow \{z_j(t)\} \subset \mathcal{M}_{\mathrm{log}}. \qquad (659) $$
### 16.2.1.1. Gaussian-weighted Modal Amplification of the Radial Equation

We amplify the radial Eq. (649) by introducing the weighted self-adjoint formulation, spectral problem and the modal amplitude equations.

Let

$$ \mathcal{J}_w(r) = r^{n-1}e^{-r^2/4} \qquad (660) $$

be the radial Gaussian-type weight, and

$$ \langle \varphi, g \rangle_{\mathcal{J}_w} = \int_0^\infty \varphi(r)g(r)\mathcal{J}_w(r)dr \qquad (661) $$

the weighted inner product.

The linear operator can be written as

$$ \Delta_{\xi}|_{\text{radial}} - \frac{1}{2}\xi \cdot \nabla_{\xi} = \frac{1}{\mathcal{J}_w(r)} \partial_r (\mathcal{J}_w(r)\partial_r(\cdot)), \qquad (662) $$

where $\Delta$ is for the full Fokker-Planck (Ornstein-Uhlenbeck) operator in divergence form.

The linearized radial operator about the profile $\varphi(r)$ is

$$ (S_L)_r[\eta](r) = \eta_{rr} + \frac{n-1}{r}\eta_r - \frac{1}{2}r\eta_r + e^{\varphi(r)}\eta = \frac{1}{\mathcal{J}_w(r)}\partial_r(\mathcal{J}_w(r)\partial_r\eta) + e^{\varphi(r)}\eta. \qquad (663) $$

The spectral problem for radial eigenmodes reads

$$ (S_L)_r[\Phi_j] = \lambda_j \Phi_j, \qquad (664) $$

where the eigenfunction $\Phi_j$ associated with $\lambda_j$ lies in the weighted space

$$ L^2((0, \infty); \mathcal{J}_w(r)dr), \qquad (665) $$

regular at $r = 0$ and integrable as $r \to \infty$ with the weight $\mathcal{J}_w(r)$.

The Rayleigh quotient associated to an eigenpair $(\lambda_j, \Phi_j)$ is

$$ \lambda_j = \frac{\int_0^\infty (-|\Phi'_j(r)|^2 + e^{\varphi(r)}|\Phi_j(r)|^2) \mathcal{J}_w(r) dr}{\int_0^\infty |\Phi_j(r)|^2 \mathcal{J}_w(r) dr}. \qquad (666) $$

For a perturbation $\eta(r, \tau)$ with $v(\xi, \tau) = \varphi(r) + \eta(r, \tau)$, the non-linear evolution is

$$ \partial_{\tau} \eta = (S_L)_r[\eta] + \text{NL}(\eta), \qquad (667) $$

in which the non-linear term is

$$ \text{NL}(\eta) = e^{\varphi(r)}(e^{\eta} - 1 - \eta) = \sum_{m \ge 2} \frac{e^{\varphi(r)}}{m!} \eta^m. \qquad (668) $$

Let $A_j(\tau)$ denotes the modal amplitudes. We decompose the perturbation as a superposition of modes

$$ \eta(r, \tau) = \sum_j A_j(\tau)\Phi_j(r). \qquad (669) $$

The amplitudes satisfy the system of ODEs

$$ \dot{A}_j(\tau) = \lambda_j A_j(\tau) + \frac{1}{2} \sum_{p,q} c_{j;pq} A_p(\tau) A_q(\tau) + \frac{1}{6} \sum_{p,q,k} d_{j;pqk} A_p(\tau) A_q(\tau) A_k(\tau) + \mathcal{O}(\|A\|^4), \qquad (670) $$

with the asymptotic $\mathcal{O}$-notation.

The quadratic interaction coefficients are

$$ c_{j;pq} = \frac{\int_0^\infty e^{\varphi(r)} \Phi_j(r) \Phi_p(r) \Phi_q(r) \mathcal{J}_w(r) dr}{\int_0^\infty \Phi_j(r)^2 \mathcal{J}_w(r) dr}, \qquad (671) $$
and the cubic interaction coefficients are

$$d_{j;pqk} = \frac{\int_{0}^{\infty} e^{\varphi(r)} \Phi_{j}(r) \Phi_{p}(r) \Phi_{q}(r) \Phi_{k}(r) \zeta_{w}(r) dr}{\int_{0}^{\infty} \Phi_{j}(r)^{2} \zeta_{w}(r) dr} \quad (672)$$

### 16.2.1.2. Exact algebraic reduction and Hamiltonian structure

Starting from Eq. (649) it is useful to introduce the standard exponential substitution and a further radial gauge which leads to an algebraic reduction. Let

$$y(r) = e^{\frac{\varphi(r)}{2}}, \quad \varphi = 2 \ln y, \quad (673)$$

and

$$\varphi_r = 2 \frac{y_r}{y}, \quad \varphi_{rr} = 2 \left( \frac{y_{rr}}{y} - \frac{y_r^2}{y^2} \right). \quad (674)$$

Substituting into the radial equation and simplifying gives the exact second-order equation for $y$,

$$y_{rr} = \frac{y_r^2}{y} - \frac{n-1}{r} y_r + \frac{r}{2} y_r - \frac{1}{2} y^3 + y. \quad (675)$$

Set

$$z = \frac{r^2}{4} \Rightarrow r = 2\sqrt{z}, \quad u(z) = ry(r), \quad y(r) = \frac{u(z)}{r}. \quad (676)$$

Via chain rules,

$$\frac{d}{dr} = \frac{r}{2} \frac{d}{dz}, \quad \frac{d^2}{dr^2} = \frac{1}{2} \frac{d}{dz} + \frac{r^2}{4} \frac{d^2}{dz^2} = \frac{1}{2} \frac{d}{dz} + z \frac{d^2}{dz^2}, \quad (677)$$

one computes explicitly that (675) is algebraically equivalent to the following ODE for $u(z)$,

$$u_{zz} = \frac{u_z^2}{u} + \left(1 - \frac{n}{2z}\right) u_z + \left(\frac{1}{z} + \frac{n-2}{4z^2}\right) u - \frac{1}{8z^2} u^3. \quad (678)$$

Eq. (678) can be conveniently rewritten as a first-order system with the logarithmic derivative,

$$w(z) = \frac{u_z}{u}. \quad (679)$$

From which this equivalent system follows,

$$\left\{ \begin{array}{l} u_z = uw, \\ w_z = -\left(1 - \frac{n}{2z}\right)w + \frac{1}{z} + \frac{n-2}{4z^2} - \frac{1}{8z^2}u^2. \end{array} \right. \quad (680)$$

In deriving (680), we used the identity $w_z = \frac{u_{zz}}{u} - \left(\frac{u_z}{u}\right)^2$, which expresses $w_z$ directly in terms of $u$ and its derivatives.

We therefore pass the canonical conjugate variable $p$ by the linear relation,

$$p(z) = u_z - \frac{1}{2}u + \mathcal{F}_{(g)}(z)u, \quad (681)$$

for a scalar gauge function $\mathcal{F}_{(g)}(z) = 1 - \frac{n}{4z}$ to simplify poles at $z = 0$.

Then there exists a rational Hamiltonian $\mathcal{H}(u, p; z)$, polynomial in $p$ and rational in $u, z$, such that the $z$-evolution is Hamiltonian,

$$\left\{ \begin{array}{l} u_z = \frac{\partial \mathcal{H}}{\partial p}, \\ p_z = -\frac{\partial \mathcal{H}}{\partial u}. \end{array} \right. \quad (682)$$

One convenient explicit choice, which reproduces (678) after elimination of $p$, is

$$\mathcal{H}(u, p; z) = \frac{1}{2} p^2 + \left(1 - \frac{n}{2z}\right) \frac{up}{2} + \frac{1}{16z^2} u^4 - \frac{1}{2} \left(\frac{1}{z} + \frac{n-2}{4z^2}\right) u^2. \quad (683)$$

This confirms the reduction of the radial equation into a rational Hamiltonian system, well suited for comparison with Painlevé-type dynamics.
Scholium 16.1 (Non-local Volterra chain for the self-similar $u(z)$-profile). We deduce a Volterra-type nested integral expansion for the self-similar profile $u(z)$ starting from Eq. (678) and its equivalent first-order system (680).

Work under the condition that the chosen branch is regular, and fix a base point $z_0 > 0$ with $u(z_0) = U_0$ and $w(z_0) = W_0$. Suppose further that $u, w$ are regular on $[z_0, z]$; if the limit $z_0 \to 0^+$ is required, it must be justified by a small-$z$ regularity expansion.

Let

$$u_z = uw, \qquad (684)$$

$$w_z = -\left(1 - \frac{n}{2z}\right)w + \frac{1}{z} + \frac{n-2}{4z^2} - \frac{1}{8z^2}u^2. \qquad (685)$$

Introduce the integrating factor for the linear part of (685),

$$\mu(z) = \exp \left\{ - \int_{z_0}^{z} \left(1 - \frac{n}{2s}\right) ds \right\} = e^{-(z-z_0)} \left(\frac{z_0}{z}\right)^{\frac{n}{2}}. \qquad (686)$$

Employing $\mu$, an integral representation for $w$ is derived,

$$w(z) = \mu(z)^{-1} \left( W_0 \mu(z_0) + \int_{z_0}^{z} \mu(s) \left[ \frac{1}{s} + \frac{n-2}{4s^2} - \frac{1}{8s^2} u(s)^2 \right] ds \right). \qquad (687)$$

The profile $u$ is recovered by quadrature,

$$u(z) = U_0 \exp \left( \int_{z_0}^{z} w(s) ds \right). \qquad (688)$$

We set up a Picard-type iteration that treats the non-linear term $-\frac{1}{8s^2}u(s)^2$ as a forcing built from the previous iterate. All new iterates inherit the $z$-differentiation convention, i.e. $u_z^{(n)} = \frac{d}{dz}u^{(n)}$ and $w_z^{(n)} = \frac{d}{dz}w^{(n)}$,

$$u^{(0)}(z) = U_0, \qquad (689)$$

$$w^{(0)}(z) = \mu(z)^{-1} \left( W_0 \mu(z_0) + \int_{z_0}^{z} \mu(s) \left[ \frac{1}{s} + \frac{n-2}{4s^2} \right] ds \right), \qquad (690)$$

The recursion is

$$w^{(n+1)}(z) = \mu(z)^{-1} \left( W_0 \mu(z_0) + \int_{z_0}^{z} \mu(s) \left[ \frac{1}{s} + \frac{n-2}{4s^2} - \frac{1}{8s^2} (u^{(n)}(s))^2 \right] ds \right), \qquad (691)$$

$$u^{(n+1)}(z) = U_0 \exp \left( \int_{z_0}^{z} w^{(n+1)}(s) ds \right), \qquad (692)$$

for $n \ge 0$.

Unwinding the recursion produces an explicit nested integral. For any $k \ge 1$ one may write $u^{(k)}$ as a finite sum of $m$-fold iterated integrals; one convenient symbolic form is

$$u^{(k)}(z) = U_0 \exp \left( \int_{z_0}^{z} w^{(k)}(s) ds \right) = U_0 \sum_{m=0}^{\infty} \frac{1}{m!} \int_{z_0}^{z} \dots \int_{z_0}^{z} \prod_{j=1}^{m} w^{(k)}(t_j) dt_1 \dots dt_m, \qquad (693)$$

where each $w^{(k)}(t_j)$ is given by the integral formula (691) and thus itself contains an integral of $(u^{(k-1)})^2$.

By successive substitution one has a true $k$-fold nested integral expression where every level corresponds either to an exponential-quadrature layer, from (692), or to a linear integral layer with kernel $\mu$, from (691).

The first non-trivial layers are explicitly obtained by substituting (691) into (692) and expanding the exponential: the $m$th exponential term produces $m$ nested integrals, each of which carries inner integrals produced by the $\mu$-kernel; iterating $k$ times gives an expression whose depth grows combinatorially with $k$.
The full Picard sum is written as

$$u(z) = \lim_{k \to \infty} u^{(k)}(z) = U_0 \exp \left( \int_{z_0}^{z} w(s) ds \right), \qquad (694)$$

with $w$ given by (687).

As usual the expansion is formal unless one imposes regularity hypotheses, e.g. bounds on $U_0, W_0$ and on $u$ on the integration interval, or contraction estimates for the map $u^{(n)} \mapsto u^{(n+1)}$.

Under such hypotheses the Picard iteration converges and the nested integral chain converges to the unique regular profile $u$ selected by the boundary conditions at $z_0$, or by the regular branch at $z=0$ in the appropriate limit.

For a quantitative control on the non-linear Volterra chain, one expands the structure directly in the original variables, keeping every integral explicit,

$$u^{(2)}(z) = U_0 \exp \left( \int_{z_0}^{z} \mu(t)^{-1} \left[ W_0 \mu(z_0) + \int_{z_0}^{t} \mu(s) \left( \frac{1}{s} + \frac{n-2}{4s^2} \right) ds \right] dt \right) \quad (695)$$

$$-\frac{1}{8} U_0^2 \sum_{m=0}^{\infty} \frac{1}{m!} \int_{z_0}^{z} \mu(t)^{-1} \int_{z_0}^{t} \mu(s) \frac{1}{s^2} \int_{z_0}^{s} dr_1 \int_{z_0}^{r_1} dr_2 \cdots \int_{z_0}^{r_{m-1}} dr_m \prod_{j=1}^{m} w^{(1)}(r_j) ds dt. \quad (696)$$

To analyse convergence, one rewrites the absolute value of each $m$-fold term. Each inner integral chain satisfies

$$\left| \int_{z_0}^{s} dr_1 \int_{z_0}^{r_1} dr_2 \cdots \int_{z_0}^{r_{m-1}} dr_m \prod_{j=1}^{m} w^{(1)}(r_j) \right| \le \frac{(s-z_0)^m}{m!} \left( \sup_{r \in [z_0, s]} |w^{(1)}(r)| \right)^m, \quad (697)$$

so that the full nested contribution in the exponent is bounded by

$$\left| \int_{z_0}^{t} \mu(s) \frac{1}{s^2} \int_{z_0}^{s} \cdots \int_{z_0}^{r_{m-1}} \prod_{j=1}^{m} w^{(1)}(r_j) dr_m \cdots ds \right| \le \int_{z_0}^{t} \mu(s) \frac{1}{s^2} \frac{(s-z_0)^m}{m!} \left( \sup_{r \in [z_0, s]} |w^{(1)}(r)| \right)^m ds. \quad (698)$$

Inserting this bound into the exponential gives the majorant series,

$$\sum_{m=0}^{\infty} \frac{1}{m!} \int_{z_0}^{z} \mu(t)^{-1} \int_{z_0}^{t} \mu(s) \frac{1}{s^2} \frac{\left(2(s-z_0) \sup_{r \in [z_0, s]} |w^{(1)}(r)|\right)^m}{m!} ds dt, \quad (699)$$

whose inner power series equals,

$$\sum_{m=0}^{\infty} \frac{1}{m!} \left( 2(s-z_0) \sup_{r \in [z_0, s]} |w^{(1)}(r)| \right)^m = \exp \left( 2(s-z_0) \sup_{r \in [z_0, s]} |w^{(1)}(r)| \right), \quad (700)$$

hence the exponent of $u^{(2)}$ converges absolutely, for any $z$ such that $\sup_{r \in [z_0, z]} |w^{(1)}(r)| < \infty$.

Expanding once more the exponential in $u^{(2)}$ produces the Volterra hierarchy for higher orders. For each $k \ge 3$,

$$u^{(k)}(z) = U_0 \exp \left\{ \int_{z_0}^{z} \mu(t)^{-1} \left[ W_0 \mu(z_0) + \int_{z_0}^{t} \mu(s) \left( \frac{1}{s} + \frac{n-2}{4s^2} \right) ds \right] dt \right. \\ - \frac{1}{8} U_0^2 \sum_{m_1, \dots, m_{k-2} \ge 0} \frac{2^{m_1+\dots+m_{k-2}}}{m_1! \dots m_{k-2}!} \int_{z_0}^{z} \mu(t)^{-1} \int_{z_0}^{t} \mu(s) \frac{1}{s^2} \\ \left. \times \left( \int_{z_0}^{s} dr_1 \dots \int_{z_0}^{r_{m_1-1}} dr_{m_1} \dots \int_{z_0}^{r_{m_{k-3}-1}} dr_{m_{k-2}} \prod_{i=1}^{k-2} \prod_{j=1}^{m_i} w^{(1)}(r_j) \right) ds dt \right\}, \quad (701)$$

making the factorial decay explicit and showing how the exponential series naturally reorganizes into a convergent Volterra sum over multi-indices $(m_1, \dots, m_{k-2})$. This expression preserves the
full non-linear structure of the iteration and exhibits the precise combinatorial scaling that controls
the analytic convergence of the Picard tower.
◇

Scholium 16.2 (Canonical Hamiltonian transformations and second variation). We extend the Hamiltonian structure (683) associated to the displayed ℌ(u, p; z) by giving the Poisson structure, the Hamilton–Jacobi equation, the canonical generating function and the exact 2 × 2 Hessian appearing in the second variation.

Let

$$
\alpha = \begin{pmatrix} u \\ p \end{pmatrix}, \quad M_{\omega} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \tag{702}
$$

be the canonical vector and the standard symplectic matrix, respectively.

Let

$$
\frac{d\alpha}{dz} = M_{\omega} \nabla_{(u,p)} \mathcal{H}(u, p; z), \qquad (703)
$$

which is equivalent to the displayed system,

$$
u_z = \frac{\partial \mathcal{H}}{\partial p}, \quad p_z = -\frac{\partial \mathcal{H}}{\partial u}. \tag{704}
$$

The Poisson bracket on smooth observables $\{u, \vartheta\}$ is

$$
\{u, \vartheta\} = \nabla_{(u,p)} u \cdot M_{\omega} \nabla_{(u,p)} \vartheta. \quad (705)
$$

The Hamilton–Jacobi equation for the action $\mathcal{S}_{\text{HJ}}(u, z)$ reads

$$
\frac{\partial \mathcal{S}_{\text{HJ}}}{\partial z}(u, z) + \mathcal{H}\left(u, \frac{\partial \mathcal{S}_{\text{HJ}}}{\partial u}(u, z); z\right) = 0. \quad (706)
$$

A type-2 generating function $\mathcal{S}(u, u^*; z)$ producing a canonical transformation $(u, p) \mapsto (u^*, p^*)$
is guaranteed by

$$
p = \frac{\partial \mathcal{S}}{\partial u}(u, u^*; z), \quad p^* = - \frac{\partial \mathcal{S}}{\partial u^*}(u, u^*; z), \qquad (707)
$$

and the new Hamiltonian satisfies

$$
K_{\mathcal{H}}(u^*, p^*; z) = \mathcal{H} \left( u, \frac{\partial \mathcal{S}}{\partial u}(u, u^*; z); z \right) + \frac{\partial \mathcal{S}}{\partial z}(u, u^*; z). \quad (708)
$$

Compute the 2 × 2 Hessian of ℌ with respect to (u, p); evaluated at a trajectory (u(z), p(z))
this gives the quadratic form appearing in the second variation. And from which one has

$$
\mathcal{S}_{\text{HJ}}^{(2)}(z) = \nabla_{(u,p)}^2 \mathcal{H}(u,p;z) = \begin{pmatrix} \frac{3u^2}{4z^2} - \left(\frac{1}{z} - \frac{n-2}{4z^2}\right) & -\frac{1}{2}\left(1 - \frac{n}{2z}\right) \\ -\frac{1}{2}\left(1 - \frac{n}{2z}\right) & 1 \end{pmatrix}, \quad (709)
$$

with $u = u(z)$ and $p = p(z)$ on the trajectory.

The linearized variational equation for $\delta\alpha = (\delta u, \delta p)^{\mathrm{T}}$ is

$$
\delta \alpha_z = M_{\omega} \mathcal{S}_{\text{HJ}}^{(2)}(z) \delta \alpha. \qquad (710)
$$

Accordingly the second variation of the action functional $\mathcal{S}_{\text{HJ}}$ is

$$
\delta^2 \mathcal{S}_{\text{HJ}} = \frac{1}{2} \int \delta\alpha(z)^{\mathrm{T}} \mathcal{S}_{\text{HJ}}^{(2)}(z) \delta\alpha(z) dz, \quad (711)
$$

for the quadratic form used to test linear stability of trajectories.

◇

Scholium 16.3 (Peano–Baker chain for the variational flow). We construct an explicit nested integral expansion for the fundamental solution of the linearized variational system, to complete the previous Scholium 16.2.

Fix a base point $z_0 > 0$ and denote by $\delta\alpha(z)$ the variation vector. Let

$$
\delta \alpha_z(z) = A(z) \delta \alpha(z), \quad A(z) = M_{\omega} \mathcal{S}_{\text{HJ}}^{(2)}(z). \tag{712}
$$

The fundamental matrix $\Phi(z)$ is defined by the initial-value problem

$$
\Phi_z(z) = A(z)\Phi(z), \quad \Phi(z_0) = \mathbb{I}_{2 \times 2}, \tag{713}
$$
and the solution of the variational equation unfolds as

$$ \delta\alpha(z) = \Phi(z)\delta\alpha(z_0). \qquad (714) $$

The Peano–Baker expansion [62] [5] for $\Phi(z)$ is the following integral series,

$$ \Phi(z) = \mathbb{I}_{2 \times 2} + \sum_{m=1}^{\infty} \int_{z_0 < t_m < \dots < t_1 < z} A(t_1) \cdots A(t_m) dt_1 \cdots dt_m. \qquad (715) $$

Since $\mathcal{I}_{HJ}^{(2)}(z)$ is symmetric, one may write

$$ A(z) = M_{\omega} \mathcal{I}_{HJ}^{(2)}(z), \quad M_{\omega} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. \qquad (716) $$

Through the matrix (709),

$$ A(z) = \begin{pmatrix} S_{12}(z) & 1 \\ -S_{11}(z) & -S_{12}(z) \end{pmatrix}, \qquad (717) $$

with

$$ S_{11}(z) = \frac{3}{4z^2} u(z)^2 - \left( \frac{1}{z} + \frac{n-2}{4z^2} \right), \quad S_{12}(z) = -\frac{1}{2} \left( 1 - \frac{n}{2z} \right). \qquad (718) $$

Thus every mth term in (715) is an m-fold nested integral whose integrand is a matrix product of the form

$$ A(t_1)A(t_2)\cdots A(t_m), \qquad (719) $$

and, after inserting (717) (718), these products become explicit combinations of $u(t_j)$, powers of $t_j$, and rational factors in $z$ and $n$.

The first two terms read

$$ \Phi(z) = \mathbb{I}_{2 \times 2} + \int_{z_0}^{z} A(t_1) dt_1 + \int_{z_0}^{z} dt_1 \int_{z_0}^{t_1} dt_2 A(t_1) A(t_2) + \cdots, \qquad (720) $$

from which one reconstructs

$$ \delta u(z) = \Phi_{11}(z)\delta u(z_0) + \Phi_{12}(z)\delta p(z_0), \qquad (721) $$

$$ \delta p(z) = \Phi_{21}(z)\delta u(z_0) + \Phi_{22}(z)\delta p(z_0), \qquad (722) $$

where each $\Phi_{ij}(z)$ is itself given by the Peano–Baker integrals in (715).

A Picard-type recursion for the truncated fundamental matrix is immediate,

$$ \Phi^{(0)}(z) = \mathbb{I}_{2 \times 2}, \qquad (723) $$

$$ \Phi^{(n+1)}(z) = \mathbb{I}_{2 \times 2} + \int_{z_0}^{z} A(t) \Phi^{(n)}(t) dt, \qquad (724) $$

so that $\Phi^{(n)} \to \Phi$ as $n \to \infty$ and the recursion unwinds into the same nested integral chain displayed above.

The Peano–Baker series (715) is formal unless one imposes bounds ensuring absolute summability, e.g. an estimate such as $\sup_{t \in [z_0, z]} \|A(t)\|(z - z_0) < 1$, or, more generally, refined operator-norm bounds ensuring convergence. Under such hypotheses, the series converges absolutely and specifies the unique fundamental solution $\Phi(z)$ solving (713). ◇

### 16.2.2. Abstract Arrangement of Blow-Up Manifolds

Let $F_g$ be a complex analytic scale of ultradifferentiable Gevrey-type spaces $\mathcal{G}^{s,\delta}(\mathbb{R}^d)$ with $s > 1$, $\delta > 0$, forming a nested sequence of compact embeddings,

$$ \dots \hookrightarrow \mathcal{G}^{s_1, \delta_1}(\mathbb{R}^d) \hookrightarrow \mathcal{G}^{s_2, \delta_2}(\mathbb{R}^d) \hookrightarrow \dots \hookrightarrow F_g. \qquad (725) $$

The blow-up manifolds are modeled as subanalytic stratified subsets inside the functional scale $F_g$. Let us see what it is about.
(1) *Similarity blow-up manifold* $\mathcal{M}_{\text{sim}}$. Let $S_{\varphi}$ denote the set of all admissible self-similar profiles $\varphi$ solving

$$ -\alpha\varphi - \beta\xi \cdot \nabla_{\xi}\varphi + \text{O}_{\text{NL}}[\varphi] = 0, \qquad (726) $$

where $\text{O}_{\text{NL}}$ is a non-local operator analytic in $\varphi$, for which $\text{O}_{\text{NL}}[\varphi] = \Delta\varphi + e^{\varphi}$. Let

$$ \mathcal{M}_{\text{sim}} = \left\{ u(x, t) = (T-t)^{-\alpha} \varphi \left( \frac{x}{(T-t)^{\beta}} \right) : (\varphi, \alpha, \beta) \in S_{\varphi} \times \mathbb{R}^2 \right\}. \qquad (727) $$

So $\mathcal{M}_{\text{sim}}$ is the stable leaf of a center manifold under the renormalized flow,

$$ \mathcal{R}_{\varepsilon}[u](x, t) = \varepsilon^{\alpha} u(\varepsilon^{\beta} x, T - \varepsilon t). \qquad (728) $$

We set up a chart $\varphi_{\text{sim}}: S_{\varphi} \times \mathbb{R}^2 \to F_{\varepsilon}$,

$$ \varphi_{\text{sim}}(\varphi, \alpha, \beta) = (T - t)^{-\alpha} \varphi \left( \frac{x}{(T - t)^{\beta}} \right), \qquad (729) $$

putting $\mathcal{M}_{\text{sim}} \simeq \text{Im}(\varphi_{\text{sim}})$ as a finite-codimension real-analytic submanifold. Note that $\text{Im}(\varphi_{\text{sim}})$ is the image of the map $\varphi_{\text{sim}}$,

$$ \text{Im}(\varphi_{\text{sim}}) = \{ \varphi_{\text{sim}}(\varphi, \alpha, \beta) : (\varphi, \alpha, \beta) \in S_{\varphi} \times \mathbb{R}^2 \}. \qquad (730) $$

(2) *Log-lattice singularity manifold* $\mathcal{M}_{\text{log}}$. Let $\text{sing}_t(u)$ be the singular support of $u(\cdot, t)$ in the complexified domain $\mathbb{C}^d$. Establish

$$ \mathcal{M}_{\text{log}} = \left\{ u \in F_{\varepsilon} : \exists \{z_j(t)\} \subset \mathbb{C}^d, \text{ sing}_t(u) = \{z_j(t)\}, \lim_{t \to T^{-}} \min_j \Im(z)_j(t) = 0 \right\}. \qquad (731) $$

We say that $A(t) = \inf\{\rho_{\text{an}} > 0: u(\cdot + iy, t)$ analytic for $|y| < \rho_{\text{an}}\}$ is the analyticity radius of $u(\cdot, t)$ in the complex spatial variable; and we interpret $\mathcal{M}_{\text{log}}$ as the set of $u$ such that

$$ \lim_{t \to T^{-}} A(t) = 0. \qquad (732) $$

In spectral space,

$$ |\hat{u}_k(t)| \sim e^{-i\langle t \rangle |k|}, \quad i\langle t \rangle \to 0, \qquad (733) $$

where $i\langle t \rangle$ is the Gevrey strip width in Fourier dual.

(3) *Stratified intersection and microlocal degeneration.* We view both $\mathcal{M}_{\text{sim}}$ and $\mathcal{M}_{\text{log}}$ as stratified analytic varieties with charts in $\mathcal{G}^{s, \delta}$. Their intersection is described microlocally via blow-down morphisms,

$$ \pi_{\epsilon}: u(x, t) \mapsto v_{\epsilon}(\xi) = \epsilon^{\alpha} u(\epsilon^{\beta} \xi, T - \epsilon), \quad \epsilon \to 0^{+}. \qquad (734) $$

Let $\mathcal{F}: F_{\varepsilon} \to \mathbb{R}$ be a singular Morse–Lyapunov in which

$$ \left. \frac{d}{d\epsilon} \mathcal{F}[v_{\epsilon}] \right|_{\epsilon \to 0} = 0, \quad \left. \frac{d^2}{d\epsilon^2} \mathcal{F}[v_{\epsilon}] \right|_{\epsilon \to 0} > 0. \qquad (735) $$

The minimizers of $\mathcal{F}$ lie on $\mathcal{M}_{\text{sim}} \cap \overline{\mathcal{M}_{\text{log}}}$.

Finally, define the Fredholm index of the tangent map between the manifolds,

$$ \text{ind} \left( T_u \mathcal{M}_{\text{log}} \to \frac{T_u F_{\varepsilon}}{T_u \mathcal{M}_{\text{sim}}} \right) = \text{dim ker}(D_u) - \text{dim coker}(D_u), \qquad (736) $$

Under generic spectral condensation, this index is zero,

$$ \text{ind} = 0 \Rightarrow \mathcal{M}_{\text{sim}} \cap \overline{\mathcal{M}_{\text{log}}}, \qquad (737) $$

viz. transversal intersection with stable persistence under perturbations.
### 16.2.3. Geometric Patterns and Flow Dynamics

Let $F_\epsilon$ be a complex scale of ultradifferentiable spaces with analytic continuation across strip domains. Let us move on to introduce the renormalized evolution operator,

$$ \mathcal{Y}[u](s, \xi) = e^{\alpha s} u(\xi e^{\beta s}, T - e^{-s}), \quad s \in \mathbb{R}_{+}, \qquad (738) $$

generating a semiflow $\mathcal{Y}^s: F_\epsilon \to F_\epsilon$. Indicate with $\mathcal{M}_{\text{sim}}$ the fixed-point set of $\mathcal{Y}^s$,

$$ \mathcal{M}_{\text{sim}} = \{u \in F_{\epsilon}: \mathcal{Y}^s u = u, \forall s \in \mathbb{R}_{+}\}, \qquad (739) $$

and let

$$ \mathcal{M}_{\text{log}} = \{u \in F_{\epsilon} \mid \exists \varphi \in \mathcal{M}_{\text{sim}} \text{ such that } \lim_{s \to \infty} \mathcal{Y}^s u = \varphi \text{ in } F_{\epsilon}\}. \qquad (740) $$

Thus $\mathcal{M}_{\text{log}}$ lies on the stable manifold of $\mathcal{M}_{\text{sim}}$ under $\mathcal{Y}$.

Let $J_{x_0}^\infty F_\epsilon$ denote the $\infty$-jet space at $x_0$,

$$ J_{x_0}^{\infty} F_{\epsilon} = \left\{ \left( D^{\alpha} u(x_0) \right)_{\alpha \in \mathbb{N}^d} \right\}, \qquad (741) $$

and fix the map $\mathcal{Y}: \mathcal{M}_{\text{log}} \to J_{x_0}^\infty F_\epsilon$ assigning to each $u$ the full jet of its asymptotic rescaling limit,

$$ \mathcal{Y}(u) = \left( \lim_{s \to \infty} \partial_{\xi}^{\alpha} \mathcal{Y}[u](s, \xi) \Big|_{\xi=0} \right)_{\alpha \in \mathbb{N}^d}. \qquad (742) $$

We deduce that $\text{Im}(\mathcal{Y}) \subset J_{x_0}^\infty \mathcal{M}_{\text{sim}}$, and the dynamics of pole condensation correspond to convergence in the pro-topology induced by the jet filtration.

**Theorem 16.2** (Renormalized convergence via jet-morphism). Let $u \in \mathcal{M}_{\text{log}}$ such that $\mathcal{Y}^s u \to \varphi \in \mathcal{M}_{\text{sim}}$ in $F_\epsilon$. For all multi-indices $\alpha$,

$$ \lim_{s \to \infty} \partial_{\xi}^{\alpha} \mathcal{Y}[u](s, \xi) \Big|_{\xi=0} = \partial_{\xi}^{\alpha} \varphi(\xi) \Big|_{\xi=0}. \qquad (743) $$

That is, $\mathcal{Y}(u) = j_0^\infty \varphi$.

*Proof.* Letting $v(s, \xi) = \mathcal{Y}[u](s, \xi)$, one gets

$$ \partial_s v = O_L[v] + O_{NL}[v], \quad v(s, \xi) \to \varphi(\xi), \qquad (744) $$

with $O_L$ being a linear dilation generator,

$$ O_L[v] = \alpha v + \beta \xi \cdot \nabla_{\xi} v, \qquad (745) $$

and $O_{NL}[v] = \Delta_{\xi} v + (e^v - e^{\varphi})$ analytic in $v$.

Let $\delta(s) = \|v(s, \cdot) - \varphi\|_{\mathcal{Y}^s(s, \cdot)} \to 0$ exponentially. The convergence in the $\infty$-jet topology follows from analyticity of $v$ in $\xi$ and the Cauchy integral of derivatives,

$$ \partial_{\xi}^{\alpha} v(s, 0) = \frac{\alpha!}{(2\pi i)^d} \oint_{|\partial_j|=r} \frac{v(s, \bar{\partial})}{\bar{\partial}^{\alpha+1\times}} d\bar{\partial}_1 \cdots d\bar{\partial}_d, \quad r < \rho_{\text{an}}(s). \qquad (746) $$

where $1_\times = (1, \dots, 1)$ and $\rho_{\text{an}}(s) \to \rho_{\text{an}*} > 0$ uniformly as $s \to \infty$. Thereby, since $v(s, \xi) \to \varphi(\xi)$ uniformly in strips, we find

$$ \lim_{s \to \infty} \partial_{\xi}^{\alpha} v(s, 0) = \partial_{\xi}^{\alpha} \varphi(0). \qquad (747) $$

□

Hence, the entire manifold $\mathcal{M}_{\text{sim}}$ can be reconstructed as the asymptotic attractor under $\mathcal{Y}$ acting on $\mathcal{M}_{\text{log}}$. The renormalized flow realizes a moduli morphism,

$$ \mathcal{Y}^{\infty}: \mathcal{M}_{\text{log}} \to \mathcal{M}_{\text{sim}}, \quad u \mapsto \lim_{s \to \infty} \mathcal{Y}[u](s), \qquad (748) $$

which is smooth on open strata and continuous on closures. This gives a natural retraction map $\mathcal{Y}^{\infty}: \overline{\mathcal{M}_{\text{log}}} \to \mathcal{M}_{\text{sim}}$, identifying $\mathcal{M}_{\text{sim}}$ as a center manifold of codimension zero in $\overline{\mathcal{M}_{\text{log}}}$.

This justifies the geometric conjecture,

$$ \overline{\mathcal{M}_{\text{log}}} \supset \mathcal{M}_{\text{sim}}, \qquad (749) $$

in the strong topology of $F_\epsilon$, and dynamically,

$$ \forall u \in \mathcal{M}_{\text{log}}, \exists s_0 > 0 \text{ such that } \inf_{\varphi \in \mathcal{M}_{\text{sim}}} \| \mathcal{Y}^s u - \varphi \| = \mathcal{O}_{s \to \infty}(0), \ s \ge s_0, \qquad (750) $$
equivalently, letting $\mathbb{P}_{\mathcal{M}_{\text{sim}}} : F_g \to \mathcal{M}_{\text{sim}}$ be the nearest-point projection,

$$ \| \Psi^s u - \mathbb{P}_{\mathcal{M}_{\text{sim}}} (\Psi^s u) \| = \Theta_{s \to \infty}(0). \qquad (751) $$
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