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This study investigates aggregation equations with porous medium diffusion, revealing bifurcations and phase transitions. We classify conditions for continuous or discontinuous phase transitions, impacting the limit behavior.

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Area of Science:

  • Mathematical Physics
  • Nonlinear Partial Differential Equations
  • Statistical Mechanics

Background:

  • Aggregation equations model phenomena like pattern formation and collective behavior.
  • Degenerate diffusion, specifically of the porous medium type, introduces mathematical complexities.
  • Understanding stationary solutions and phase transitions is crucial for predicting system long-term behavior.

Purpose of the Study:

  • To analyze stationary solutions and phase transitions in aggregation equations with porous medium-type degenerate diffusion (exponent m).
  • To investigate the existence and properties of bifurcations from homogeneous steady states.
  • To characterize the nature (continuous or discontinuous) of phase transitions based on parameters m and interaction potentials W.

Main Methods:

  • Analysis of stationary solutions using bifurcation theory.
  • Investigation of free energy functionals to identify minimizers.
  • Classification of phase transition types based on mathematical criteria.
  • Asymptotic analysis for the limit m -> 1.

Main Results:

  • Existence of potentially infinite bifurcations from the spatially homogeneous steady state.
  • Proof of existence and, for weak interactions, uniqueness of free energy minimizers.
  • Classification of parameters (m, W) leading to continuous or discontinuous phase transitions.
  • Demonstration of the influence of phase transitions on the limit m -> 1.

Conclusions:

  • The study provides a comprehensive analysis of aggregation dynamics with degenerate diffusion.
  • Phase transitions play a significant role in the system's behavior and limiting properties.
  • The findings offer insights into pattern formation and collective phenomena governed by such equations.