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Global Existence and Nonlinear Stability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with Large Initial Data of Spherical Symmetry

José A. Carrillo, Samuel R. Charles, Gui-Qiang G. Chen, Difan Yuan

TL;DR

The paper addresses global existence and unconditional nonlinear stability of finite-energy, radially symmetric solutions to the multidimensional compressible Euler-Riesz equations with large data, using a vanishing-viscosity approach from the Navier-Stokes-Riesz regularization. It introduces a free-boundary CNSRE formulation to obtain uniform BD-type entropy and higher-integrability estimates, then passes to the inviscid limit to obtain global weak solutions of the Euler-Riesz system. A concentration-compactness–driven variational analysis yields existence and (up to translations) uniqueness of steady states as energy minimizers and proves their nonlinear stability under symmetric perturbations. The results cover a broad range of Riesz and logarithmic potentials, including sub- and super-Coulomb regimes, and establish a rigorous link between nonlocal interactions, steady states, and global dynamics with potential applications in astrophysical and biological aggregation models. The methodology combines radial potential representations, BD entropy, compensated compactness, and energy minimization principles to overcome nonlocality and singularity challenges inherent in CEREs.

Abstract

The compressible Euler-Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy solutions of the multidimensional Euler-Riesz equations with large initial data of spherical symmetry. We consider both attractive and repulsive interactions for a wide range of Riesz and logarithmic potentials for dimensions larger than or equal to two. This is achieved by the inviscid limit of the solutions of the corresponding Cauchy problem for the Navier-Stokes-Riesz equations. The strong convergence of the vanishing viscosity solutions is achieved through delicate uniform estimates in $L^p$. It is observed that, even if the attractive potential is super-Coulomb, no concentration is formed near the origin in the inviscid limit. Moreover, we prove that the nonlinear stability of global finite-energy solutions for the Euler-Riesz equations is unconditional under a spherically symmetric perturbation around the steady solutions. Unlike the Coulomb case where the potential can be represented locally, the singularity and regularity of the nonlocal radial Riesz potential near the origin require careful analysis, which is a crucial step. Finally, unlike the Coulomb case, a Grönwall type estimate is required to overcome the difficulty of the appearance of boundary terms in the sub-Coulomb case and the singularity of the super-Coulomb potential. Furthermore, we prove the nonlinear stability of global finite-energy solutions for the compressible Euler-Riesz equations around steady states by employing concentration compactness arguments. Steady states properties are obtained by variational arguments connecting to recent advances in aggregation-diffusion equations.

Global Existence and Nonlinear Stability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with Large Initial Data of Spherical Symmetry

TL;DR

The paper addresses global existence and unconditional nonlinear stability of finite-energy, radially symmetric solutions to the multidimensional compressible Euler-Riesz equations with large data, using a vanishing-viscosity approach from the Navier-Stokes-Riesz regularization. It introduces a free-boundary CNSRE formulation to obtain uniform BD-type entropy and higher-integrability estimates, then passes to the inviscid limit to obtain global weak solutions of the Euler-Riesz system. A concentration-compactness–driven variational analysis yields existence and (up to translations) uniqueness of steady states as energy minimizers and proves their nonlinear stability under symmetric perturbations. The results cover a broad range of Riesz and logarithmic potentials, including sub- and super-Coulomb regimes, and establish a rigorous link between nonlocal interactions, steady states, and global dynamics with potential applications in astrophysical and biological aggregation models. The methodology combines radial potential representations, BD entropy, compensated compactness, and energy minimization principles to overcome nonlocality and singularity challenges inherent in CEREs.

Abstract

The compressible Euler-Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy solutions of the multidimensional Euler-Riesz equations with large initial data of spherical symmetry. We consider both attractive and repulsive interactions for a wide range of Riesz and logarithmic potentials for dimensions larger than or equal to two. This is achieved by the inviscid limit of the solutions of the corresponding Cauchy problem for the Navier-Stokes-Riesz equations. The strong convergence of the vanishing viscosity solutions is achieved through delicate uniform estimates in . It is observed that, even if the attractive potential is super-Coulomb, no concentration is formed near the origin in the inviscid limit. Moreover, we prove that the nonlinear stability of global finite-energy solutions for the Euler-Riesz equations is unconditional under a spherically symmetric perturbation around the steady solutions. Unlike the Coulomb case where the potential can be represented locally, the singularity and regularity of the nonlocal radial Riesz potential near the origin require careful analysis, which is a crucial step. Finally, unlike the Coulomb case, a Grönwall type estimate is required to overcome the difficulty of the appearance of boundary terms in the sub-Coulomb case and the singularity of the super-Coulomb potential. Furthermore, we prove the nonlinear stability of global finite-energy solutions for the compressible Euler-Riesz equations around steady states by employing concentration compactness arguments. Steady states properties are obtained by variational arguments connecting to recent advances in aggregation-diffusion equations.

Paper Structure

This paper contains 15 sections, 41 theorems, 375 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical Problems and Construction of Approximate Solutions
  3. Weak solutions and steady states of CEREs and CNSREs
  4. Main results
  5. Approximate PDEs
  6. Uniform Estimates of the Approximate Solutions
  7. Estimates on the radial non-local potential
  8. Energy estimates
  9. Expanding the domain
  10. BD-type entropy
  11. Higher integrability
  12. Global Solutions to the Compressible Navier-Stokes-Riesz Equations
  13. Nonlinear Stability of Steady States for the Compressible Euler-Riesz Equations
  14. Convolution Inequalities
  15. Construction of the Approximate Initial Data

Key Result

Theorem 2.4

Consider problem 0.0--0.1 with $\alpha \in (-1,n-1)$ and the initial data of spherical symmetry as defined in 0.3--0.4. Assume that $(\rho_0,{\mathcal{M}}_0)$ satisfy 0.7--0.8, $\gamma > \frac{1}{n-1-\alpha}$, and Then there exists a global spherically symmetric finite-energy solution $(\rho,{\mathcal{M}})(t,\boldsymbol{x})$ of problem 0.0--0.1 in the sense of Definition definition.

Figures (1)

  • Figure 1: Relation between $\alpha$ and $n$.

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Existence of Spherically Symmetric Solutions of CEREs
  • Remark 2.5
  • Theorem 2.6: Nonlinear Stability of Steady States for CEREs
  • Remark 2.7
  • Remark 2.8
  • Lemma 3.1: Continuity of the Potential
  • proof
  • ...and 63 more