Entropic solutions to the 1D pressureless Euler system with nonlocal interactions

Technical Summary: Entropic Solutions to the 1D Pressureless Euler System with Nonlocal Interactions

Problem Statement
This work investigates the global well-posedness of weak solutions for the one-dimensional pressureless Euler-Poisson-alignment (EPA) system. The system models collective dynamics where particles interact via three mechanisms: long-range attraction, short-range repulsion (modeled by a Poisson force), and mid-range alignment. The governing equations are:
$$\begin{cases} \partial_t\rho + \partial_x(\rho u) = 0, \\ \partial_t(\rho u) + \partial_x(\rho u^2) = -\rho \partial_x \phi + \int_{\mathbb{R}} \phi(x-y)(u(y)-u(x))\rho(x)\rho(y)dy, \\ -\partial_{xx}^2 \phi = 2\kappa\rho, \end{cases}$$
where $\kappa$ determines the nature of the Poisson interaction ($\kappa < 0$ for attraction, $\kappa > 0$ for repulsion). While global smooth solutions exist for subcritical initial data, supercritical data lead to finite-time singularity formation. The primary challenge addressed is the non-uniqueness of distributional weak solutions after singularity formation. The authors seek a selection principle to identify a unique, physically meaningful solution, specifically examining the compatibility of such solutions with "sticky particle" dynamics (where colliding particles merge and move together).

Methodology
The authors employ a reduction strategy that transforms the coupled system into a scalar balance law with a time-dependent flux.

1. Scalar Reformulation: By introducing auxiliary quantities related to the cumulative distribution function $M(x,t) = \int_{-\infty}^x \rho(y,t)dy$ and a primitive of the momentum, the EPA system is reduced to the scalar equation:
   $$\partial_t M + \partial_x (A(M, t)) = (\phi * M) \partial_x M,$$
   where the flux $A(M, t)$ is explicitly time-dependent due to the Poisson term: $A(m, t) = A(m) + \kappa t m^2$. The term $(\phi * M)$ represents the nonlocal alignment force.
2. Entropy Selection Principle: To resolve non-uniqueness, the authors define an entropy solution for this scalar balance law using Kruzkov entropy pairs $(\eta, q)$ satisfying specific entropy inequalities. This framework extends previous work on the Euler-alignment system (where $\kappa=0$) to accommodate the time-dependent flux induced by the Poisson force.
3. Discrete Approximation: For the attractive case ($\kappa < 0$), the authors construct solutions via a discrete "sticky particle" approximation. They define a system of $N$ particles obeying ODEs with Cucker-Smale alignment and Poisson forces, supplemented by collision rules where particles merge inelastically (conserving momentum and sticking together).
4. Limit Passage: Using Helly's selection theorem and uniform bounds derived from the discrete dynamics, the authors pass to the limit as $N \to \infty$ to establish the existence of entropy solutions for general initial data in the attractive regime.

Key Contributions and Results

• Global Well-Posedness for Entropy Solutions: The paper establishes the existence, uniqueness, and $L^1$-stability of entropy solutions for the scalar balance law (4) with time-dependent flux. The stability estimate is given by:
  $$\|M(\cdot, t) - \tilde{M}(\cdot, t)\|_{L^1} \leq \|M_0 - \tilde{M}_0\|_{L^1} + t \|A - \tilde{A}\|_{Lip}.$$
  This result guarantees that the selection principle yields a unique weak solution for the original EPA system.

• Regime-Dependent Behavior:

  • Attractive Regime ($\kappa < 0$): The authors prove that the unique entropy solution is compatible with sticky particle dynamics. Specifically, the discrete sticky particle solutions converge to the entropy solution of the scalar balance law. In this regime, the Oleinik entropy condition is satisfied by the sticky particle collisions, ensuring that mass concentrations (shocks) persist and evolve according to the inelastic collision rules.
  • Repulsive Regime ($\kappa > 0$): A fundamental qualitative difference is revealed. The authors demonstrate that the sticky particle solution is not an entropy solution in the repulsive case. The Oleinik entropy condition fails for discrete sticky solutions because the repulsive force causes atomic states (point masses) to instantaneously disperse into absolutely continuous states (rarefaction waves). Consequently, the entropy solution in the repulsive regime does not support the "sticking" behavior; instead, it describes the spreading of mass.

• Analytical Refinements: The work addresses new analytical difficulties arising from the coexistence of a genuinely nonlocal term (alignment) and an explicitly time-dependent flux (Poisson). The authors refine entropy admissibility conditions and stability estimates, showing that while the time-dependence of the flux does not affect the stability bound's structure, it requires careful handling in the doubling-of-variables technique used for uniqueness proofs.

Significance
The paper claims to provide a rigorous mathematical framework for continuing pressureless Euler systems with nonlocal interactions beyond singularity formation. Its primary significance lies in:

1. Uniqueness: Establishing a unique selection principle (entropy solutions) for the EPA system, resolving the ambiguity inherent in distributional weak solutions.
2. Physical Insight: Highlighting a dichotomy between attractive and repulsive regimes. In the attractive case, the macroscopic entropy solution aligns with the microscopic sticky particle model, validating the use of sticky dynamics for modeling aggregation. In the repulsive case, the entropy solution predicts the dispersion of singularities, indicating that sticky particle dynamics are insufficient to describe the physical evolution when repulsion dominates.
3. Methodological Extension: Successfully extending the scalar balance law approach, previously applied to Euler-alignment and Euler-Poisson systems separately, to the coupled EPA system with time-dependent fluxes.

The authors note that while the entropy solution is mathematically well-posed, the physical relevance of the repulsive case solution (which allows for the "un-sticking" or dispersion of mass) remains a subject of debate compared to alternative weak solutions that might permit different behaviors.