Stationary Nontrivial Homogeneous Solutions

• Stationary nontrivial homogeneous solutions are time-independent profiles in nonlinear PDEs that maintain spatial variability and scaling properties.
• They emerge as global energy minimizers below a critical threshold where strong aggregation overcomes diffusive dispersion.
• Numerical simulations, spectral analysis, and uniqueness proofs validate their stability, symmetry, and compact support in applications like pattern formation.

A stationary nontrivial homogeneous solution is a time-independent solution of a nonlinear (often nonlocal) PDE or system that is both spatially nonconstant and preserves a particular homogeneity or scaling property. Such solutions are central in the analysis of pattern formation, singularity models, and long-range interaction phenomena in nonlinear diffusion, kinetic theory, fluid dynamics, and related areas. In many contexts, stationary nontrivial homogeneous solutions arise as minimizers or critical points of variational principles, serve as attractors for long-time dynamics, or illuminate phase transitions associated with competing forces (e.g., diffusion vs. aggregation).

1. Mathematical Formulation and Energetics

A typical framework is furnished by the one-dimensional nonlocal aggregation-diffusion equation studied in (Burger et al., 2011): $$\frac{\partial \rho}{\partial t} = \nabla\cdot\left[\rho\, \nabla(\rho - G*\rho)\right]$$ where $\rho(x,t)$ is a density on $\mathbb{R}^d$, $G$ is a nonnegative, radial, smooth, attractive, integrable kernel, and $*$ denotes convolution. The stationary problem is: $$0 = \nabla \cdot \left[ \rho \nabla(\rho - G*\rho) \right]$$ or, when integrated over each (possibly disconnected) component of the support of $\rho$: $$\rho(x) = \int_{\mathrm{supp}\,\rho} G(x-y)\,\rho(y)\,dy + C$$ with $C$ a constant related to the value of the energy functional at $\rho$.

The evolution equation is the $L^2$-Wasserstein gradient flow of the energy functional: $$E[\rho] = \frac{\varepsilon}{2} \int_{\mathbb{R}^d} \rho^2(x)\,dx - \frac{1}{2} \iint_{\mathbb{R}^d\times\mathbb{R}^d} G(x-y)\rho(x)\rho(y)\,dx\,dy$$ where the quadratic diffusion term encodes local repulsion and the (negative) interaction term encodes long-range attraction.

Critical points (including stationary nontrivial homogeneous solutions with prescribed mass) satisfy

$\rho \nabla(\rho - G*\rho) = 0 \quad \text{a.e.}$

It follows that $\rho = G*\rho + C$ on connected components of the support.

2. Existence and Threshold Phenomenon

The existence of stationary nontrivial homogeneous solutions is governed by a parameter threshold. With normalization $\|G\|_1 = 1$:

• If $\varepsilon \geq 1$, no nontrivial steady state exists; all initial data disperse and the only stationary solution is $\rho \equiv 0$.
• If $\varepsilon < 1$, a nontrivial, compactly supported, and symmetric steady state exists, which globally minimizes $E[\rho]$ among all densities with prescribed mass.

The threshold $\varepsilon_c = 1$ is sharp. In the critical case, there is no nonzero $L^2$ stationary solution, and rigorous asymptotic and spectral analysis excludes the existence of nontrivial steady states at this point. The presence of the threshold is a manifestation of the competition between quadratic diffusion and attractive nonlocal aggregation: stationary pattern formation emerges only if the attractive interaction is sufficiently strong to overcome diffusion.

3. Uniqueness, Regularity, and Symmetry (1D Case)

In one dimension, stationary nontrivial homogeneous solutions exhibit:

• Uniqueness: Up to translation and mass normalization, the stationary state is unique if $G$ is strictly decreasing on $(0,\infty)$.
• Symmetry: Any nontrivial steady state is symmetric about its center of mass, i.e., $\rho(x_0 - x) = \rho(x_0 + x)$, and strictly decreasing away from the center.
• Compact Support: Solutions are $C^2$ on their support, which is a single interval $[x_0-L, x_0+L]$.
• Monotonicity: The profile is strictly decreasing on $(x_0, x_0+L)$ and strictly increasing on $(x_0-L, x_0)$.
• Concavity: For sufficiently small $\varepsilon$ (strong attraction), solutions are strictly concave.
• Global Energy Minimizers: Nontrivial steady states minimize $E[\rho]$ globally among nonnegative densities of prescribed mass.

These properties are established via reduction to a symmetric eigenproblem for compact, strongly positive integral operators and the application of a strong version of the Krein–Rutman theorem. Rearrangement inequalities and spectral gap arguments enforce uniqueness and symmetry.

4. Computation and Numerical Simulation

Explicit analytical solutions are usually unavailable except in special cases (e.g., exponential or Gaussian kernels with certain parameters). The qualitative profile is accessed via:

• Finite-difference discretization of the stationary equation over a finite interval, with compact support imposed by thresholding.
• Explicit Euler schemes for dynamic evolution, which show convergence to the steady profile for $\varepsilon < 1$.
• Spectral calculation of the principal eigenvalue of the relevant compact operator (arising from linearization around the steady state), verifying monotonicity with respect to the support length and the tightness of the threshold at $\varepsilon = 1$.
• Kernels: Both smooth (e.g., Gaussian, $G(x)=e^{-x^2/2}$) and singular (e.g., exponential $G(x)=\frac{1}{2}e^{-|x|}$) interaction kernels confirm the qualitative scenarios predicted by the theory.

5. Biological and Physical Interpretation

These mathematical results translate into precise predictions for natural and engineered systems exhibiting aggregation:

• For population dynamics and swarming models, the system models individuals with short-range repulsion (diffusion) and long-range attraction (aggregation), such as chemotactic bacteria, animal groups, or synthetic agents.
• The threshold phenomenon quantifies the critical balance required for spatially localized population clusters (patterns) to emerge.
• The uniqueness and robustness of the profile in 1D imply that the aggregate structure is stable and predictable for parameter values below the threshold.
• Compact support and monotonicity correspond to a core group with high density tapering off rapidly—a feature widely observed in biological aggregations.

6. Energy Landscape and Gradient Flows

Stationary nontrivial homogeneous solutions correspond to global minimizers of an energy landscape featuring both local and nonlocal terms. These solutions are attractors for the gradient flow with respect to the Wasserstein metric, indicating:

• Dynamic stability: Small perturbations converge back to the steady state under the prescribed evolution.
• Gradient structure: The underlying evolution is interpretable as steepest descent with respect to $E[\rho]$, unifying dissipative mechanisms (diffusion) with conserved or competitive mechanisms (interaction).
• Stationarity Relations: On their support, steady states satisfy $\rho - G*\rho = \mathrm{const.}$ and are characterized by Lagrange multipliers encoding mass and center-of-mass constraints.

7. Outlook and Generalizations

The results admit several natural generalizations:

• Higher dimensions: Many results extend with suitable modifications, but uniqueness and explicit characterization become substantially harder due to the lack of order structure and convexity.
• Other interaction kernels: Extensions include kernels with singularities, anisotropy, or hard-core constraints. The general threshold phenomenon and variational structure persist, but regularity and profile shape may vary.
• Nonquadratic diffusion: More general nonlinear or degenerate diffusive dynamics introduce new thresholds and pattern types, relevant in porous media and crowd motion models.
• Kinetic and mean-field systems: Analogous stationary nontrivial homogeneous solutions arise in kinetic frameworks (e.g., mean-field Fokker–Planck, kinetic aggregation models), often characterized via fixed-point or self-consistency equations.

The analytical scenario described in (Burger et al., 2011)—balance of local and nonlocal terms, a sharp threshold, and robust symmetric profiles—serves as a prototype for homogeneous stationary aggregation patterns in a wide range of nonlinear, nonlocal PDEs from mathematical biology, physics, and social science.