The nonlocal attraction-repulsion transport equation with power kernels

Technical Summary: The Nonlocal Attraction-Repulsion Transport Equation with Power Kernels

Problem Statement
This paper investigates the nonlocal continuity equation on $\mathbb{R}^d$ describing the evolution of a probability density $\phi_t$ driven by the competition between attraction toward a prescribed background measure $\omega$ and self-repulsion among particles. The dynamics are governed by the equation:
$$\partial_t \phi_t = \text{div}\left(\phi_t (\nabla V - K_r * \phi_t)\right), \quad t > 0,$$
where the background potential is $V = \psi_a * \omega$, and the interaction kernels are power laws defined by $\psi_s(x) = |x|^{1+s}$ and $K_s(x) = \nabla \psi_s(x) = (1+s)|x|^{s-1}x$. The exponents $a$ (attraction) and $r$ (repulsion) lie in the range $[0, 1)$.

The study focuses on the "discordant" regime where $a \neq r$, as well as the equal-exponent case $a=r$ with arbitrary mass $\omega(\mathbb{R}^d)$. The equation is formally the Wasserstein gradient flow of the interaction energy:
$$E(\phi) = \int_{\mathbb{R}^d} (\psi_a * \omega) d\phi - \frac{1}{2} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \psi_r(x-y) d\phi(x) d\phi(y).$$
When $a=r$ and $\omega$ is a probability measure, this energy corresponds to the Maximum Mean Discrepancy (MMD) with a negative-distance kernel. The primary mathematical challenges arise from the singularity of the kernels at the origin (for $s=0$) and the lack of global Lipschitz bounds, which prevents the direct application of standard gradient flow theories requiring $\lambda$-convexity.

Methodology
The authors employ a multi-faceted analytical approach:

1. Squared-Radius Regularization: To handle the singularities and lack of Lipschitz continuity, the authors introduce a regularization $\psi_{s,\epsilon}(x) = (|x|^2 + \epsilon)^{(1+s)/2}$. Unlike convolution mollification, this preserves convexity and one-sided sign estimates exactly. They establish global well-posedness for the regularized system and pass to the limit $\epsilon \to 0$ using compactness arguments.
2. Lagrangian Framework: Solutions are constructed and analyzed as push-forwards of the initial data by a flow map $X_t$, satisfying the characteristic ODE $\dot{X}_t = -\nabla V(X_t) + K_r * \phi_t(X_t)$.
3. Free-Boundary Characterization: For stationary states, the authors utilize Fourier analysis and the theory of homogeneous distributions. They invert the convolution operators using fractional Laplacians to reduce the stationarity condition $\phi \nabla F_\phi = 0$ to a free-boundary problem involving a fractional exterior Dirichlet problem.
4. Energy and Moment Estimates: Uniform $L^\infty$ bounds and moment estimates are derived via Gronwall-type arguments on the regularized system. These are crucial for proving the tightness of solutions and the confinement of the support in attractive regimes.

Key Contributions and Results

• Global Well-Posedness: The paper establishes global Lagrangian well-posedness for arbitrary initial data in $L^1 \cap L^\infty$ with finite moments. The solution is unique within the class of Lagrangian distributional solutions satisfying uniform $L^\infty$ and moment bounds. Furthermore, the authors prove the propagation of $W^{n,\infty}$ regularity for smooth initial data.
• Uniform Compactness of Support: In the attractive-dominant regimes ($a > r$, or $a=r$ with $\omega(\mathbb{R}^d) > 1$), the authors prove that if the initial data is compactly supported, the support of the solution remains uniformly bounded for all time. This is achieved by showing that the diameter of the support cannot grow indefinitely due to the dominance of the attractive force over repulsion at large distances. A counterexample is provided showing this property fails if $a=r > 1$.
• Characterization of Stationary States: For the non-quadratic attractive-dominant range ($0 \le r \le a < 1$), the paper characterizes zero-flux stationary states as solutions to a free-boundary problem. Specifically, the stationary density $\phi$ is expressed as $\phi = Q_{a,r} - L_r F$, where $L_r$ is an inverse convolution operator related to the fractional Laplacian, and $F$ satisfies a fractional exterior Dirichlet problem.
• Explicit Stationary Examples: Using the derived characterization, the authors construct explicit stationary measures for specific geometries:
  • 1D: Interval solutions involving Riesz potentials.
  • 2D: Disk solutions utilizing the Boggio-Kelvin Green kernel for the fractional Laplacian.
  • 3D: Solutions for uniform ball and Gaussian backgrounds, where the characterization reduces to local differential operators (Laplacian) due to the specific exponents ($a=r=0$).
• Long-Time Convergence: The paper proves that every global solution with bounded energy and uniform moment bounds possesses a subsequence converging to a zero-flux stationary state. If the $\omega$-limit set contains a unique stationary state, full convergence is established.

Significance and Claims
The authors position this work as an extension of previous one-dimensional and equal-exponent results to arbitrary dimensions $d \ge 1$ and the discordant regime $a \neq r$.

• Theoretical Advancement: The paper provides a rigorous framework for analyzing singular attraction-repulsion flows where the energy is not $\lambda$-convex, a condition that typically precludes standard gradient flow existence theories. The use of squared-radius regularization is highlighted as a key technical innovation that preserves necessary structural properties.
• Modeling Insight: The results offer a prototype model for "unequal access" in resource distribution. In the attractive-dominant case, the equilibrium population $\phi_\infty$ is attracted to the resource distribution $\omega$ but confined by internal repulsion, resulting in a stationary state that does not coincide with $\omega$. The explicit characterization of $\phi_\infty$ allows for the quantification of inequality in wealth/resource distribution.
• Connection to MMD: The paper clarifies the relationship between these transport equations and the Maximum Mean Discrepancy (MMD) used in machine learning. While previous works addressed the MMD gradient flow for smooth kernels or specific cases, this paper extends the analysis to the singular negative-distance kernel in higher dimensions, establishing well-posedness and asymptotic behavior for the case $a=r$ with $\omega(\mathbb{R}^d) \ge 1$.
• Numerical Validation: The theoretical stationary profiles derived from the free-boundary characterization are verified against numerical particle simulations, showing agreement in both one and two dimensions.

The authors explicitly note that their results on uniform support confinement do not apply to the case $a=r$ with $\omega(\mathbb{R}^d)=1$ if $\omega$ is not compactly supported, as the support is expected to grow to match $\omega$. They also acknowledge that the case $a=r$ with $\omega(\mathbb{R}^d)=1$ is being addressed in parallel work by Rosenzweig, Slepčev, and Wang, focusing on mean-field convergence rates.