Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High-friction limit for bipolar Euler-Riesz systems

Nuno J. Alves, Jan Haskovec

TL;DR

The paper analyzes a two-species Euler--Riesz system and rigorously justifies the high-friction limit, showing convergence to a bipolar aggregation-diffusion system with Riesz interactions. Using a relative entropy framework, it establishes a stability inequality that compares dissipative weak solutions of the scaled system with strong solutions of the limiting system, under suitable admissibility conditions on the adiabatic exponents and a small coupling. The main result demonstrates that the relative energy remains controlled and converges to zero as the relaxation parameter vanishes, effectively proving a weak-strong convergence in the high-friction regime. The analysis extends prior results for single-species Euler--Riesz and bipolar Euler--Poisson systems to the more general bipolar Riesz setting and highlights the role of nonlocal interactions and gamma-dependent regularity in two-species coupling.

Abstract

We consider a bipolar Euler-Riesz system and rigorously justify the high-friction limit of weak solutions towards a bipolar aggregation-diffusion system with Riesz interactions. The analysis is carried out via the relative entropy method in the regime where smooth solutions of the limiting equations exist. This extends previous results on the high-friction limit of bipolar Euler-Poisson systems to a more general class of interactions, and extends the one-species Euler-Riesz case to the bipolar setting.

High-friction limit for bipolar Euler-Riesz systems

TL;DR

The paper analyzes a two-species Euler--Riesz system and rigorously justifies the high-friction limit, showing convergence to a bipolar aggregation-diffusion system with Riesz interactions. Using a relative entropy framework, it establishes a stability inequality that compares dissipative weak solutions of the scaled system with strong solutions of the limiting system, under suitable admissibility conditions on the adiabatic exponents and a small coupling. The main result demonstrates that the relative energy remains controlled and converges to zero as the relaxation parameter vanishes, effectively proving a weak-strong convergence in the high-friction regime. The analysis extends prior results for single-species Euler--Riesz and bipolar Euler--Poisson systems to the more general bipolar Riesz setting and highlights the role of nonlocal interactions and gamma-dependent regularity in two-species coupling.

Abstract

We consider a bipolar Euler-Riesz system and rigorously justify the high-friction limit of weak solutions towards a bipolar aggregation-diffusion system with Riesz interactions. The analysis is carried out via the relative entropy method in the regime where smooth solutions of the limiting equations exist. This extends previous results on the high-friction limit of bipolar Euler-Poisson systems to a more general class of interactions, and extends the one-species Euler-Riesz case to the bipolar setting.

Paper Structure

This paper contains 7 sections, 5 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Energy and relative energy
  3. Weak solutions to the bipolar Euler--Riesz system
  4. Statement of the main result
  5. Convergence in the high-friction limit
  6. Results on the internal energy
  7. Proof of Theorem \ref{['thm_main']}

Key Result

Proposition 3.2

Let $d > 1$ and $1 < \alpha < d$. If $f,g \in C([0,T); L^\gamma(\Omega) )$ with $\gamma \geq 2d/(d+\alpha)$, then $K_\alpha \ast f \in C([0,T); L^{\frac{2d}{d-\alpha}}(\Omega) )$ and $g (K_\alpha \ast f) \in C([0,T); L^{1}(\Omega) )$. Moreover, the weak spatial gradient of $K_\alpha \ast f$ is given and belongs to $C([0,T); L^{\frac{2d}{d-\alpha+2}}(\Omega) )$. Therefore, where $c = d-\alpha -1$.

Theorems & Definitions (8)

  • Definition 3.1
  • Proposition 3.2
  • Theorem 4.1
  • Remark 4.2
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • proof