arXiv subjects

Samuel R. Charles

Publications and source records attributed to Samuel R. Charles.

3 recordsLinked to original sources

arXiv2026

Nonlinear stability and instability of rotating Riesz star solutions of the compressible Euler-Riesz equations

The compressible Euler--Riesz equations arise in the modelling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. We study rotating steady states of the attractive compressible Euler--Riesz equations, which we call rotating Riesz stars, and establish existence and nonlinear stability or instability results in both the mass-subcritical and mass-supercritical regimes. In the mass-subcritical regime, we prove the existence of rotating Riesz stars under suitable subhomogeneity assumptions on the angular momentum profile. We then establish their nonlinear stability through a concentration compactness argument adapted to the axisymmetric setting. Rotation creates new compactness difficulties, most notably the possibility that minimising sequences are tight along rings whose radii diverge to infinity. In the mass-supercritical regime, we prove existence in the polytropic setting under suitable superhomogeneity assumptions on the angular momentum profile, thereby extending the theory beyond the small angular velocity regime. The proof requires a careful analysis of mass-preserving scalings, which are more delicate than in the non-rotating case. Finally, by analysing the concavity of the free-energy along these scalings, we establish the instability of the resulting mass-supercritical rotating Riesz stars. Our results show that rotation can have either a stabilising or a destabilising effect, depending on the singularity of the Riesz interaction.

math.AP↗

arXiv2026

Nonlinear Stability and Instability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with General Pressure Laws

The compressible Euler-Riesz equations arise in the modeling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. In this paper, we investigate the nonlinear stability and instability of steady states for the multidimensional compressible Euler-Riesz equations under general pressure laws. In the polytropic case, we establish the nonlinear instability of steady states in the mass-supercritical regime for attractive potentials; this is achieved by analyzing the concavity of the free energy along mass-preserving dilations. At the mass-critical exponent, we show that, for any steady state, there exist solutions that start arbitrarily close to it, but develop growing support. For general pressure laws, we employ a concentration-compactness approach to prove the existence of energy minimizers and establish the nonlinear stability of steady states. Moreover, we quantify the finite-time stability by deriving a relative entropy bound for finite-energy solutions, without requiring uniform pointwise upper and lower bounds on the density. We further exploit the convexity of the second moment to obtain quantitative growth estimates for solutions with positive energy, thereby proving the local nature of the stability result. Finally, we prove the global existence of finite-energy weak solutions to the compressible Euler-Riesz equations with spherical symmetry for general pressure laws via the compensated compactness method, thereby yielding unconditional stability around steady states within the class of weak solutions. The approach developed in this paper should be useful for solving other nonlinear partial differential equations involving similar difficulties.

math.AP↗

arXiv2025

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

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.

math.AP↗