Singular Homogeneous Global Solutions

• Singular homogeneous global solutions are scale-invariant PDE solutions that exhibit isolated, non-removable singularities with precise blow-up profiles.
• They are classified by asymptotic behaviors, with explicit cases in fluid dynamics such as Navier–Stokes and nonlinear elliptic equations.
• Their construction via stereographic projection and nonlinear eigenvalue problems provides benchmarks for stability, bifurcation, and regularity analyses.

Singular homogeneous global solutions are distinguished, explicitly or implicitly scale-invariant solutions of PDEs, typically arising as models for isolated singularities, critical points of variational problems, or as fundamental solutions. These solutions are "homogeneous" in the sense of possessing an exact scaling symmetry and are "singular" due to non-removability or explicit blow-up at isolated points, rays, or boundaries. Their rigorous construction, classification, and significance span linear and nonlinear elliptic equations, fluid dynamics (notably the stationary Navier–Stokes equations), free boundary problems, and subelliptic or fully nonlinear contexts.

1. Definitions and General Framework

Let $u$ solve an equation in an $n$-dimensional space (Euclidean, manifold, or a group structure) exhibiting invariance under a one-parameter group of dilations: $u(\lambda x) = \lambda^{-\alpha} u(x)$ for $\lambda > 0$, with associated degree $\alpha \in \mathbb{R}$. Solutions that preserve this structure and possess singularities at points or along lower-dimensional subsets are termed "singular homogeneous global solutions".

In the context of the stationary Navier–Stokes system in $\mathbb{R}^3$:

$$-\Delta u + (u\cdot\nabla)u + \nabla p = 0, \quad \nabla\cdot u = 0,$$

a $(-1)$-homogeneous solution has the explicit scaling $u(x) = |x|^{-1} U(\hat x)$ and $p(x) = |x|^{-2} P(\hat x)$, with $U,P$ defined on $\mathbb{S}^2$ and $\hat x = x/|x|$ (Li et al., 2024). In fully nonlinear elliptic equations,

$F(D^2u, Du, x) = 0$

on cones or domains with conical points, homogeneous solutions take the form $u(x) = |x|^{-\alpha} \varphi(x/|x|)$, governed by principal exponents $\alpha$ determined by nonlinear eigenvalue problems (Armstrong et al., 2011).

2. Removable and Non-removable Singularities in Navier–Stokes

For $(-1)$-homogeneous solutions of stationary Navier–Stokes, the removability of a singularity at a ray (corresponding to $P \in \mathbb{S}^2$) is characterized by the local behavior of $u$ near $P$:

• If $u = o(\ln\text{dist}(x,P))$ as $x\to P$ along $\mathbb{S}^2$, then the singularity is removable.
• This condition is optimal: for any $\alpha>0$, there exists $u$ such that $\lim_{x\to P}|u(x)|/\ln|x'| = -\alpha$, rendering the singularity non-removable (Li et al., 2024).

Such results are proven by reducing the problem to nonlinear elliptic equations on $\mathbb{S}^2$, leveraging the interplay between local regularity and the asymptotic expansion of profiles in spherical coordinates.

3. Classification of Isolated Singular Behaviors

Axisymmetric $(-1)$-homogeneous solutions (with or without swirl) admit a full classification according to asymptotic behavior near a singularity. The limit $T := \lim_{x\to P} |x'| u_\varphi(x)$ at $P$ determines the nature of the singularity, yielding five mutually exclusive regimes:

1. $T \ge 3$: $u_\varphi$ constant, profiles are real-analytic.
2. $2 < T < 3$: Nontrivial scaling, $u_\varphi$ diverges as $|x'|^{-1}$.
3. $T=2$: Logarithmic-type expansion with possible $\ln|x'|$ layers.
4. $0 < T < 2$: Power-law with subcritical blow-up, finite limits.
5. $T=0$: More delicate expansions involving $|x'|^{-1}\ln|x'|$ layers (Li et al., 2024, Li et al., 2016).

Correspondingly, three "types" of singularities are identified:

• Type 1 (Landau): $u$ is bounded — realized by classical Landau solutions.
• Type 2 (Logarithmic): $u \sim \alpha \ln \mathrm{dist}_{\mathbb{S}^2}$, $0 < |\alpha| < \infty$.
• Type 3 (Power-type): $u \sim \mathrm{dist}^{-\beta}$ for $0 < \beta < 1$ or higher.

4. Global Construction and Multiplicity

Using explicit representation via stereographic projection and meromorphic functions (Liouville-type formulas), it is possible to construct global $(-1)$-homogeneous solutions in all of $\mathbb{R}^3 \setminus \{0\}$ with finitely many prescribed singular rays:

• For any $m \ge 2$ distinct $P_1, ..., P_m \in \mathbb{S}^2$ and integer weights $\ell_j \notin \{0, \pm1\}$ satisfying $\sum_j \ell_j = m-2$, there exists a $u$ smooth away from $P_j$, with prescribed leading singularity at $P_j$ (Li et al., 2024).
• This approach generalizes previously known axisymmetric and non-axisymmetric families (Landau, Serrin, Liouville family) and shows the abundance and flexibility of singular homogeneous global solutions.

Example/Method	Structure on $\mathbb{S}^2$	Singular Set
Landau solutions	Smooth, axisymmetric, bounded	none/point at $0$
Serrin vortices	Axisymmetric, power/log type	$x_3$-axis/$P$
Liouville-family	Via meromorphic $f(z)$	Arbitrary finite set

5. Singular Homogeneous Solutions in Other Contexts

Fully Nonlinear Elliptic Equations

For positively homogeneous fully nonlinear equations $F(D^2u,Du,x)=0$ in a cone $C_\omega$:

• Precisely two homogeneous singular solutions $\Psi^+$ (degree $-\alpha^+$) and $\Psi^-$ (degree $-\alpha^-$) exist, unique up to scaling (Armstrong et al., 2011).
• These "model" all possible isolated boundary and interior singularities in Lipschitz domains with conical points: any nonnegative solution either vanishes continuously at the singularity or is a multiple of $\Psi^+$.
• The exponents $\alpha^+$ ($>0$) and $\alpha^-$ ($<0$) are characterized via nonlinear eigenvalue problems on the sphere (Armstrong et al., 2011).

Subelliptic and Non-Euclidean Geometries

On the Heisenberg group and for Hörmander operators:

• The fundamental solution (Green’s function) for the sublaplacian exhibits singular homogeneous behavior dictated by the homogeneous dimension $Q$ (Afeltra, 2020, Biagi et al., 2019).
• On $\mathbb{H}^n$, the Yamabe singular solution is $u(x,t) = C \rho(x,t)^{-n}$, with a genuine non-removable isolated singularity at the origin (Afeltra, 2020).
• For Hörmander operators, the global fundamental solution scales as $d(x,y)^{2-q}$ in dimension $n>2$ ($d$ Carnot–Carathéodory distance), matching the homogeneous scaling of the operator (Biagi et al., 2019).

6. Stability and Perturbation Theory

Certain singular homogeneous global solutions (notably mild log-type blow-up, Type 2 for Navier–Stokes) are not only explicit but exhibit stability properties:

• Type 2 $(-1)$-homogeneous axisymmetric no-swirl solutions of Navier–Stokes, other than Landau, are asymptotically stable under $L^3$-perturbations, with sharp $L^q$ decay rates computable explicitly.
• The analysis uses anisotropic Caffarelli–Kohn–Nirenberg inequalities, weighted Riesz transforms, and energy methods anchored in the precise singularity profile (Zhao et al., 2023).

7. Broader Synthesis and Applications

Singular homogeneous global solutions serve as:

• Model singularities: Governing local behavior near blow-up or at admissible singular points/rays for nonlinear equations.
• Classification benchmarks: Providing rigidity, uniqueness, or multiplicity results; dictating which singularities are "generic" or "removable".
• Explicit counterexamples: Illustrating sharpness of regularity, removability, or ill-posedness thresholds.
• Test fields: For stability/instability, bifurcation theory, or as energy minimizers/maximizers in variational settings.

Connections span classical potential theory, fluid mechanics, geometric PDEs, and qualitative theory for nonlinear elliptic and parabolic problems. Their explicit nature grounds theory, and their sharp classification supports both theoretical and applied developments across analysis and mathematical physics.